Enumaration of the number of spanning trees in some special planar maps

(1)

N◦ d’ordre 2587 THÈSE DE DOCTORAT

Présentée par

Abdulhaﬁdh MODABISH Discipline : Mathématiques et Informatique

Spécialité : Mathématiques Appliquées

ENUMERATION OF THE NUMBER OF SPANNING TREES

IN SOME SPECIAL PLANAR MAPS

Soutenue le 09 Juillet 2012 Devant le jury

Président :

Pr. Driss ABOUTAJDINE - Professeur à la Faculté des sciences, Rabat. Examinateurs :

Pr. El Mamoun SOUIDI - Professeur à la Faculté des Sciences, Rabat. Pr. Mohamed EL MARRAKI - Professeur à la Faculté des Sciences, Rabat. Pr. Abdelmalek AZIZI - Professeur à la Faculté des Sciences, Oujda. Pr. Hussain BEN-AZZA - Professeur Assistant à l’ENSAM, Mèknas.

Pr. Mohamed EL KAMILI - Professeur Assistant à la Faculté des Sciences, Fès.

Faculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat - Maroc Tel +212 (0) 5 37 77 18 34/35/38, Fax : +212 (0) 5 37 77 42 61, http://www.fsr.ac.ma

(2)

(3)

To my daughter Hanin.

(4)

(5)

ACKNOWLEDGEMENTS

The work presented in this thesis was performed at the Laboratory of MIA "Mathematics, Computer Science and Applications" - UFR of Mathematics, Computer Science and Applications, Department of Mathematics and Computer Science at the Faculty of Sciences Rabat - University of Mohammed V-Agdal.

First of all, thanks to ALLAH who awarded me with patienace, courage and self consitancy in order to successfully carry out this research work. I would like to warmly thank my worthy of respect supervisor, Mr. Mohamed El MARRAKI, Professor of Higher Education in the Faculty of Sciences Rabat, for his support, his availability, patience, close collaboration and kind assistance that allowed me to complete this thesis. I express my profound gratitude to Mr. Driss ABOUTAJDINE, Professor of Higher Education at the Faculty of Sciences Rabat, for the honor that he gave me to evaluate this work, on one hand, and to chair the thesis committee, on the other hand. Therefore, he ﬁnds the expression of my deep gratitude here.

I wish to thank Mr. El Mamoun SOUIDI, Professor of Higher Education at the Faculty of Sciences Rabat, and Mr. Abdelmalek AZIZI, professor of Higher Education at the Faculty of Sciences - University of Mohammed I - Oujda, for having accepted the burden to be the penal secretaries of thesis committee and for their valuable comments that helped me to improve this manuscript.

I am also quite grateful to Professor Hussain BEN-AZZA, University of Mollay Ismal, ENSAM in Meknes, for giving me an honor to accept the evaluation burden of my thesis being thesis examiner and for his kind help on spanning trees and for patiently answering my several questions. Very simply, thanks a lot Sir.

I extend my thanks to Mr. Mohamed EL KAMILI, Professor in the Faculty of Science - University of Sidi Mohammed Ben Abdellah - Fes by agreeing his consent to participate in the examination committee as its member and also for the discussions we had together.

Finally, my warmest thanks go to my family for always having faith in me; who gave me non conditional support and encouragement throughout my studies. Also, thanks to

(6)

my wife who has been very supportive during the period of my research. At last but not least, I remain unable to conclude without mentioning the kind assistance and thanking all the members of our research group "Laboratory of Mathematics, Computer Science and Applications" at the Faculty of sciences Rabat. My thanks of course to all my dears, I came into contact during these four years and who helped me directly or indirectly during the course of this thesis.

(7)

List of Figures

1.1 An example of graph G . . . . 22

1.2 Some examples of graph . . . 22

1.3 Example of graph with adjacent vertices . . . 23

1.4 An example of vertex degrees . . . 24

1.5 An example of vertex degrees . . . 24

1.6 An example of complete graph K5 . . . 25

1.7 From left to right, the graphs K4, K2,2, P4, C4. . . 26

1.8 A graph G and subgraphs of G. . . . 26

1.9 Some r-regular graphs. . . . 27

1.10 Complete Graphs. . . 28

1.11 Null Graphs. . . 28

1.12 Cycle Graphs. . . 28

1.13 Two graphs G and H are not the same, but they are isomorphic. . . . 29

1.14 Two graphs G and H are not isomorphic. . . . 29

1.15 The graph K4 drawn as a plane graph without edge crossing. . . 30

1.16 One graph gives two planar maps . . . 31

1.17 (a) A representation of a graph; its set of vertices is {1, 2, 3, 4}, and (multi)set of edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Two embeddings of this graph in the sphere, which are not homeomorphic since the second has a triangular face, unlike the ﬁrst. . . 31

1.18 The degree of the faces of this planar map are written inside the faces . . . 32

1.19 An example of mapC . . . 33

1.20 Path graphs. . . 34

1.21 Simple trees . . . 35

1.22 Path and Star trees . . . 35

1.23 A graph G with its 3 spanning trees . . . . 38

1.24 An example of a graph G whose diameter is 2 . . . . 39

2.1 A graph(left) and all spanning trees (right) of this graph. . . 42

2.2 A graph G gives rise to ﬁve spanning trees . . . . 43

2.3 A graph G and three of its spanning trees . . . 43

2.4 A spanning tree of graph G . . . 44

2.5 A spanning tree of graph G . . . 44 9

(12)

2.6 A graph G and its all spanning trees . . . . 46

2.7 A graph G with its adjacency matrix. . . . 47

2.8 A graph with 6 vertices. . . 47

2.9 P4 and its incidence matrix. . . 48

2.10 The incidence matrix of a graph G . . . 49

2.11 The incidence matrix M of a directed graph G . . . 49

3.1 A map C gives rise to eight spanning trees . . . 56

3.2 Graph G and its reduced graphs G1 and G2 . . . 56

3.3 The 16 diﬀerent spanning trees in K4. . . 57

3.4 A simple example graph . . . 59

3.5 A graph G and its graph oriented . . . . 60

3.6 A graph G and its adjacency matrix, degree matrix and Laplacian matrix . 61 3.7 Example graph, with Laplacian matrix and eigenvalues. Numbers near each vertex indicate the chosen ordering. The total number of spanning trees can be seen to be 8 by inspection, which matches with Kirchhoﬀ’s theorem. . . 64

3.8 The 2nd example graph, with Laplacian matrix and eigenvalues. Numbers near each vertex indicate the chosen ordering. The total number of span-ning trees can be seen to be 3 by inspection, which matches with Kirchhoﬀ’s theorem. . . 65

3.9 The Petersen graph. . . 65

4.1 A map C and its complexity which has ﬁve spanning trees . . . 68

4.2 Example of a map C which has two faces (Cycle) . . . 68

4.3 A map C = C1• C2 . . . 69

4.4 Star map and chain map . . . 70

4.5 An example of maps C, C − e and C.e . . . 70

4.6 A Contraction-Deletion method . . . 71

4.7 Computing of τ (C) by deletion-contraction. . . 72

4.8 Recursive calculation of τ (C) . . . 73

4.9 An example of maps C, C − p and C.p . . . 74

4.10 A mapC= C1 :C2 . . . 76

4.11 An example of map C= C1 : C2 . . . 76

4.12 An example of maps C= C1 : C2, C1,C2, C1.v1v2 and C2.v1v2 . . . 77

4.13 Case of Particular maps . . . 78

4.14 A mapC= C1 | C2 . . . 78

4.15 An example of maps C, C1,C2, C1− e, C2 − e, C1.e and C2.e . . . . 78

4.16 The map C after the tranformation . . . 79

4.17 An example of a map C of type C = C1 | C2 . . . 79

4.18 A mapC of type C= C1 | C2 with calculation of τ (C) . . . 80

4.19 A mapC of type C= C1 | C2 with calculation of τ (C) . . . 80

(13)

LIST OF FIGURES 11

4.21 An example of a mapC of type C = C1 ‡ C2 . . . 81

4.22 A mapC of type C= C1 ‡ C2 with calculation of τ (C) . . . 82

4.23 A mapC of type C= C1 ‡ C2 with calculation of τ (C) . . . 82

5.1 Example of A map C which has two faces (Cycle) . . . 86

5.2 Case of two cycles . . . 86

5.3 A simple planar map has 23 spanning trees . . . 87

5.4 Case of n cycles . . . 87

5.5 Case of n cycles whose lengths are the same with hi = h and ki = k . . . . 88

5.6 Case of n cycles whose lengths are the same with hi = h and ki = 1 . . . . 89

5.7 The n-Fan chains Fn . . . 90

5.8 The n-Grid chains Gn. . . 91

5.9 The n-Tent chains Tn . . . 91

5.10 The n-Hexagonal chains Hn . . . 92

5.11 The n-Eight chains En . . . 92

5.12 The n-Home chains Hn . . . 94

5.13 The n-Barrel chains Bn . . . 95

5.14 The n-Light chains Ln . . . 96

5.15 Particular case of maps Cn and Qn . . . 98

5.16 The n-Kite chains Kn . . . 100

5.17 A planar map 2-Kite chains and its Laplacian matrix . . . 101

5.18 The n-Envelope chains En . . . 103

5.19 The n-Diphenylene chains Dn . . . 105

6.1 A star ﬂower planar graph (map) . . . 108

6.2 A cycle Cn and the star ﬂower planar map . . . 109

6.3 The star ﬂower planar map Sn,k . . . 109

6.4 The star ﬂower planar maps S2,k and S3,k . . . 110

6.5 The star ﬂower planar map Sn,k after cutting . . . 110

6.6 The star ﬂower planar map Sn,1 . . . 111

6.7 The star ﬂower planar maps S2,1 and S3,1 . . . 112

6.8 The star ﬂower planar map Sn,1 after cutting . . . 112

6.9 The star ﬂower planar map Sn,2 . . . 113

7.1 The utilities graph can be drawn in many ways. . . 116

7.2 Two diagrams look similar, but they are not the same graph. . . 116

7.3 Two graphs G and H are not the same, but they are isomorphic. . . 117

7.4 Two graphs both have ﬁve vertices and six edges, but are not isomorphic. . 117

7.5 The graphs G and H are not the same, however, they are isomorphic. . . . 118

7.6 Example of a mapC5 . . . 119

7.7 The maps E5,E6 and En . . . 121

7.8 The maps E3 and E4 . . . 121

7.9 The principal matrix of En . . . 124

(14)

7.11 The maps Fn and Gn . . . 129 7.12 The crystal planar map Cn . . . 131 7.13 The mapsEn, En− e and En.e . . . 132

(15)

List of Tables

5.1 Some values of τ (Fn), τ (Gn), τ (Tn), τ (Hn) and τ (En) . . . . 93

5.2 Some values of τ (Hn) . . . 94

5.3 Some values of τ (Bn) . . . 96

5.4 Some values of τ (Ln) . . . . 97

5.5 Some values of τ (Cn) and τ (Qn) . . . 99

5.6 Some values of τ (Kn) and τ (Qn) . . . 102

5.7 Some values of τ (En) and τ (Qn) . . . 104

5.8 Some values of τ (Dn) and τ (Qn) . . . 106

6.1 Some values of τ (Sn,1) . . . 113

6.2 Some values of τ (Sn,2) . . . 114

7.1 Some values of τ (En) . . . 129

7.2 Some values of τ (Fn) and τ (Gn) . . . 131

7.3 Some values of τ (Cn) . . . 132

(16)

(17)

Introduction

The branch of mathematics that studies graphs is called graph theory. A graph is a mathematical object that consists of two sorts of elements: vertices and edges. Every edge corresponds to a pair of vertices, in each case the respective two vertices are said to be adjacent. Not every pair of vertices need to be adjacent. More formally; a graph G is a triple consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints. A loop is an edge whose endpoints are equal. Multiple edges are the edges having the same pair of endpoints. A graph G is connected if each pair of vertices in G belongs to a path; otherwise, G is disconnected. A graph which contains neither multiple edges nor loops is called a simple graph. A simple path is either an edge or a path p = v0, v1, v2, ..., vn−1, vn such that deg(vi) = 2 for i = 1, 2, ..., n− 1. A cycle is a path such that v0 = vn. A graph

G is planar if it can be drawn (design) in the plane such that no edges are crossing each

other. A graph G is finite if its vertex set and edge set are finite. We adopt the convention that every graph mentioned in this thesis is finite, unless explicitly constructed otherwise. The degree of a vertex v (or valency) in a graph G, denoted by deg(v) is the number of edges incident to it (any loop incident to the vertex is counted twice). A subgraph of a graph G is a graph all of whose vertices are vertices of G and all of whose edges are edges of G. A map C is a graph drawn on a surface X or embedded into it (that is, a compact variety orientable 2-dimensional). A planar map is a map drawn on the plane. A tree is a connected graph without cycle. A plan tree is a tree designed in the plane. A spanning tree in G is a subgraph of G that includes every vertex and is also a tree. All maps in this thesis are undirected, connected, planar and without loops, because the loops don’t affect spanning trees, so we delete them before the computation.

The main object studied in this thesis is a spanning tree and calculating the number of spanning trees in planar maps (graphs embedded in the surface without edge-crossings). In a graph (network) that contains several cycles, we must remove the redundancies in this network, i.e., we obtain a spanning tree. A spanning tree in a map C is a tree which has the same vertex set as C (tree that passing through all the vertices of the map C). The number of spanning trees of a map C, also known as the complexity of C denoted by τ (C), is the total number of distinct spanning subgraphs of C that are trees. It is an important well-studied quantity, an invariant of the graph (network) and an important measure of reliability of a network. It also plays a crucial role in Kirchhoﬀ’s classical theory of electrical networks, for example in computing driving point resistance [112],

(18)

and in random walks in graphs [114]. The research of the number of spanning trees in a graph has a long history. The cornerstone of the research, the Matrix Tree Theorem, dated back to 1847, which is attributed to Kirchhoff. The Matrix Tree Theorem is a famous and classic result on the study of τ (G). It is known that Kirchhoff Matrix Tree Theorem [81, 85], can be applied to any map C to determine τ(C) by taking the determinant of Laplacian matrix L ofC, i.e., all cofactors of L are equal, and their common value is τ(C), but this requires evaluation of a determinant of a corresponding characteristic matrix. However, for a few special families of graphs there exist simple formulae which make it much easier to calculate and determine the number of corresponding spanning trees especially when these numbers are very large. One of the first such results is due to Cayley [25] who showed that complete graph on n vertices, Kn, has nn−2 spanning trees [65], that is he showed τ (Kn) = nn−2 for n ≥ 2. Another combinatorial method of counting the number of spanning trees in a graph G is the Feussner’s recursive formula [49], for counting τ (G) in a graph G, is quite intuitive. For an undirected simple graph G, let e be any edge of G. All spanning trees in G can be separated into two parts: one part contains all spanning trees without e as a tree edge; another contains all spanning trees with e as a tree edge. The first part has the same number of spanning trees as graph G1,

where G1 is the graph G with e deleted; the second part has the same number of spanning

trees as graph G2, where G2 is the graph created from G by shrinking the two vertices of e

into one vertex. Both G1 and G2 have fewer edges than G, so the number of spanning trees

in G can be counted recursively in this way. It is then clear that τ (G) = τ (G1) + τ (G2).

In this thesis, we have generalized this formula by replacing the edge e by a simple path

p that contains k edges, see [89]. Let G be a graph with multiple edges and self-loops.

When we count τ (G), we ﬁrst neglect all self-loops in G because they have no contribution to any spanning tree. If G itself is a tree then τ (G) = 1, and if G is disconnected, then

τ (G) = 0.

Our research theme in this thesis focuses on the counting of the number of spanning trees in connected planar maps, a subject in combinatorial graph theory; and to ﬁnd new methods to calculate the number of spanning trees in any map (network). Most research about the number of spanning trees is devoted to determining exact formulae for the number of spanning trees in many kinds of special graphs, see [6, 14, 62, 63, 65, 66, 127]. In this thesis, we start by stating the general methods for counting the number of spanning trees in graphs; we then provide our new results.

Spanning trees are relevant to various aspects of graphs (networks). Generally, the number of spanning trees in a network can be obtained by computing a related deter-minant of the Laplacian matrix L of the network. However, for a large map (network), evaluating the relevant determinant is computationally intractable. In this thesis, we give new methods to facilitate the calculation of the number of spanning trees in planar maps and prove novel simpliﬁed results. Then, we apply these methods on certain planar maps to derive several explicit simple formulae for calculating the number of spanning trees in some special families of planar maps which are called the n-Fan chains, the n- Grid chains, the n-tent chains, the n-Hexagonal chains, the n-Eight chains, the n-Home chains,

(19)

LIST OF TABLES 17

the n-Barrel chains, the n-Light chains, the n-Kite chains, the n-Envelope chains and the

n-Diphenylene, ... etc.

Outline of the thesis

This thesis is organized as follows. In the first chapter, some general graph theory is described. In the second chapter, we introduce the background and research history of our problem and definitions and general properties of basic objects studied (spanning trees, complexity, matrices associated to a graph, ... etc). Some basic results are also introduced. The third chapter introduces the background and research of the calculation of the number of spanning trees in graphs and some methods for counting spanning trees are introduced. In this chapter, we firstly state the general methods for counting the number of spanning trees in graphs, and then our new results are discussed. In the fourth chapter, we provide new non-trivial methods for calculating the number of spanning trees in planar maps in general then in particular in some special connected planar maps and prove new simplified results, we then apply these methods on some specials planar maps to give explicit simple formulae to calculate the number of spanning trees in certain families of planar maps in chapter five. In the sixth chapter, we consider the star flower planar map, and derive a simple explicit formulae for counting the number of spanning tree in it. in this chapter we use one of the methods which mentioned in the fourth chapter to find the number of spanning tree in the star flower planar map. Finally in the last chapter, we are going to focus on the maximal planar maps. This chapter will be divided into two sections; we devote the first section for calculating the Weiner index (the sum of distances between all pairs of vertices) in the case of planar maps in general then in particular in the maximal planar maps and in the other section shall study how to count the number of spanning trees in this type of this maps, as well as enumeration of spanning trees in the maximal planar maps. For that, we develop a method, by using the Matrix Tree Theorem as the base and manipulating the matrices, to prove that the number of spanning trees in a maximal planar map which satisfies a recurrence relation. This recurrence relation can be determined exactly, i.e., we reach at a formula to calculate the number of spanning trees in the maximal planar maps. In the end, some possible future works are proposed.

New results

The new results obtained in the framework of this thesis are concentrated in Chapters 4, 5, 6 and 7.

The results of Chapter 4 have been partially published in [89]. This paper is the cornerstone of the research subject in this thesis, as it includes new methods to calculate the number of spanning trees in connected planar maps.

The results of Chapter 5 are published in the articles [85, 87, 88, 89]. In these articles, we have applied our new methods that we have reached in the article [89] on certain

(20)

special panar maps. We were able to obtain explicit simple formulae for calculating the number of spanning trees in some special planar maps.

The results of Chapter 6 are published in article [86]. In this paper, we have been interested in calculating the number of spanning trees in the star ﬂower planar map and have derived the explicit formula to calculate the number of spanning trees in the star ﬂower planar maps.

The results of Chapter 7 were divided into two parts. The ﬁrst part emphasizes upon the determination of Weiner index in the case of planar maps, in general, and in maximal planar maps particularly, as can be seen in [90]. While, the second part focuses on the derivation of an explicit formula to calculate the number of spanning trees in a maximal planar map by employing the Laplacian matrix of planar maps (Matrix Tree Theorem) which has already been published in [84].

(21)

Part I

Preliminaries

(22)

(23)

Chapter 1 Deﬁnitions and Properties

This chapter includes a brief introduction to graph theory. Some of the basics concerning this theory are presented in [19] and [119]. In case of rare terminologies, the reader is referred to consult [69] and [117].

1.1 Graphs

In this section, we are going to present some useful deﬁnitions related to our work as follows. A graph will usually be denoted by a capital letter G.

Deﬁnition 1.1.1 An undirected graph G is a triplet (VG,EG,δ) where VG is the set of vertices of the graph G, EG is the set of edges of the graph G (EG ⊆ V2, meaning that an edge e∈ EG is a 2-element subset {vi; vj} of VG) and δ is the application:

δ : EG → P(VG)

ei 7→ δ(ei) = {vj, vk}

with vj and vk are end vertices of the edge ei (not necessarily distinct). When u and v are the endpoints of an edge, they are neighbors. We notice that the set {vj, vk} as a multiset (if vj = vk, the same vertex appears twice in δ(ei)). A loop is an edge ei ∈ EG with vj = vk (an edge whose endpoints are equal), if δ(ei) = δ(ej) with i ̸= j then the edges ei and ej are called multiple edges (edges having the same pair of endpoints). Example 1.1.2 In the graph shown in Figure 1.1, We have: EG ={e1, e2, e3, e4}, VG =

{v1, v2, v3}, δ(e1) = {v1, v1} (multiset) δ(e2) = {v1, v2} and δ(e3) = δ(e4) = {v2, v3}, the

edges e3 and e4 are multiple edges, the edge e1 is a loop, then the graph G admits a loop

and two multiple edges.

In a graph G, a path is a sequence of vertices and edges p = v0, e1, v1, e2, ..., vn−1, en, vn such that δ(ei) = {vi−1, vi}. We say that this path attached both ends v0 and vn. A cycle is a path such that v0 = vn. A graph G is ﬁnite if its vertex set and edge set are ﬁnite.

(24)

Figure 1.1: An example of graph G

We adopt the convention that every graph mentioned in this thesis is ﬁnite, un-less explicitly constructed otherwise.

Remark 1.1.3 For brevity we denote the edge {u, v} ∈ EG simply by uv. In particular when uv∈ EG we say that the vertices u and v are adjacent. Then the degree of a vertex

v, written deg(v), is the number of vertices of G which are adjacent to v. When there is

no ambiguity, we sometimes write V (G) and E(G) instead of VG and EG, respectively. In a graph, as already explained, two or more edges joining the same pair of vertices are multiple edges. An edge joining a vertex to itself is a loop. A graph with no multiple edges or loops is a simple graph. For example, graph (a) below (see Figure 1.2) has multiple edges and graph (b) has a loop; therefore, it is not a simple graph. Graph (c) has no multiple edges or loops, and is therefore a simple graph.

Figure 1.2: Some examples of graph

Deﬁnition 1.1.4

The vertices v and u of a graph are adjacent vertices if they are joined by an edge e. The

vertices v and u are incident with the edge e, and the edge e is incident with the vertices

v and u.

Example 1.1.5 In the graph below (see Figure 1.3), the vertices u and x are adjacent, vertex w is incident with edges 2, 3, 4 and 5, and edge 6 is incident with the vertex x.

(25)

1.1. Graphs 23

Figure 1.3: Example of graph with adjacent vertices

Definition 1.1.6 A graph is finite if its vertex set and edge set are finite. We adopt the convention that every graph mentioned in this thesis is finite, unless explicitly constructed otherwise.

Deﬁnition 1.1.7 We say that a graph G is connected if any two of its vertices may be connected by a path (if each pair of vertices in G belongs to a path); otherwise, G is disconnected [69, 84].

Remark 1.1.8 If G has a u, v-path (Tow vertices u and v connected by a path in G), then u is connected to v in G.

The connection relation on V (G) consists of the ordered pairs (u, v) such that u is connected to v. "Connected" is an adjective we apply only to graphs and to pairs of vertices (we never say "v is disconnected" when v is a vertex).

For a graph G, and a set S of vertices and edges in G, we shall denote G− S as the graph obtained by removing the elements of S from G. Removing a vertex from a graph, is deﬁned as removing the vertex, and all edges incident to it.

Deﬁnition 1.1.9 A graph G is said to be k-connected, if the removal of any (k − 1) vertices will not disconnect G, but there exist a set S of k vertices such that G− S is disconnected.

The graphs that we consider are in most cases connected but may contain multiple edges.

1.1.1 Vertex Degrees

In many applications of graph theory we need a term for the number of edges meeting at a vertex. For example, we may wish to specify the number of roads meeting at a particular intersection, the number of wires meeting at a given terminal of an electrical network, or the number of chemical bonds joining a given atom to its neighbors. Such situations are illustrated below (see Figure 1.4):

(26)

Figure 1.4: An example of vertex degrees

In chemistry, the term valency is used to indicate the number of bonds connecting an atom to its neighbors. For example, a carbon atom C (in excited state) has valency 4, an oxygen atom O has valency 2, and a hydrogen atom H has valency 1, as illustrated in the above diagram (see Figure. 1.4) representing the ethanol molecule. For graphs, we usually use the word degree.

Deﬁnition 1.1.10 The degree of a vertex v (or valency) in a graph G, denoted by deg(v) is the number of edges incident to it (any loop incident to the vertex is counted twice because it has two ends joined to that vertex); the degrees of the vertices are fundamental parameters of a graph. A vertex of degree 0 is isolated.

Example 1.1.11 Consider the following graphs:

Figure 1.5: An example of vertex degrees

Graph (a) has vertex degrees: deg u = 2, deg v = 1, deg w = 4, deg x = 3, deg y = 0, and graph (b) has vertex degrees: deg u = 2, deg v = 5, deg w = 4, deg x = 5, deg y = 0.

(27)

1.1. Graphs 25

The ﬁrst such problem is to count the edges; we do this using the vertex degrees. The resulting formula is an essential tool of graph theory, sometimes called the "First Theorem of Graph Theory" or the "Handshaking Lemma".

Each edge is incident to two vertices; hence:

Proposition 1.1.12 (Degree-Sum Formula) If G is a graph or multigraph, then the sum of the degrees of all vertices of G is equal to twice the number of its edges (in particular the degree sum is an even number) i.e.,

∑ v∈VG

deg(v) = 2|EG|.

Proof: Summing the degrees counts each edge twice, since each edge has two ends and

contributes to the degree at each endpoint.

Example 1.1.13 In the graph shown in Figure. 1.5 (a), we have: the number of the edges is 5 , and the sum of the degrees of all vertices of this graph is 10, and in Figure. 1.5 (b), the number of the edges is 8, the sum of the degrees of all vertices of this graph is 16.

Deﬁnition 1.1.14 A simple path is either an edge or a path p = v0, v1, v2, ..., vn−1, vn such that deg(vi) = 2 for i = 1, 2, ..., n− 1.

Deﬁnition 1.1.15 A complete graph is a graph which has one edge between each pair of distinct vertices or whose vertices are pairwise adjacent. The complete graph with n vertices is denoted by Kn; (see Figure. 1.6).

Figure 1.6: An example of complete graph K5

We shall identify four graphs which are of special characteristics.

1. A complete graph Kn is a graph with n vertices all of which are adjacent one to another. In particular K3 is also called a triangle.

2. Km,n is a graph whose vertices are partitioned into two sets of m and n elements such that two vertices are adjacent if and only if they belong to diﬀerent sets. A graph with this property is called complete bipartite.

(28)

3. A path Pn is a graph with vertices {v1, ..., vn} and edges {v1v2, v2v3, ..., vn−1vn}. It can also be called a path from v1 to vn.

4. For n ≥ 3, Cn is the graph Pn with one additional edge: vnv1. Consequently it is

called a closed path or a cycle.

Figure 1.7: From left to right, the graphs K4, K2,2, P4, C4.

In mathematics, we often study complicated objects by looking at simpler objects of the same type contained in them - subsets of sets, subgroups of groups, and so on. In graph theory we make the following deﬁnition.

Deﬁnition 1.1.16 A subgraph of a graph G is a graph, such that, all of whose vertices are vertices of G and all of whose edges are edges of G, i.e., a graph H is a subgraph of another graph G if VH ⊆ VG and EH ⊆ EG and the assignment of endpoints to edges in

H is the same as in G. We denote this relation by H ⊆ G and say that G contains H.

For example, P3 ⊆ C3 and K2,2 ⊆ K2,4.

Remark 1.1.17 Note that G is a subgraph of itself.

Example 1.1.18 The following graphs (see Figure 1.8) are subgraphs of the graph G on the left, with vertices {u, v, w, x} and edges {1, 2, 3, 4, 5}.

Figure 1.8: A graph G and subgraphs of G.

A graph in which all the vertex degrees are the same is given a special name. Deﬁnition 1.1.19 A graph G is regular if its vertices all have the same degree.

(29)

1.1. Graphs 27

Deﬁnition 1.1.20 A graph G is said to be r-regular, or regular of degree r, or simply regular, if every vertex in G has degree r, that is, every vertex has r edges incident to it. For example K3 is 2-regular and Kn is n− 1-regular. A regular 3-regular graph is called cubic. A cubic graph has an even number of vertices.

Example 1.1.21 In the following graphs, we illustrate some r-regular graphs, for various values of r:

Figure 1.9: Some r-regular graphs.

A useful consequence of the proposition (Degree-Sum Formula) 1.1.12 is the following result.

Theorem 1.1.22 Let G be an r-regular graph with n vertices. Then G has nr/2 edges. Proof: Let G be a graph with n vertices, each of degree r; then the sum of the degrees of all the vertices is nr. By the Proposition (Degree-Sum Formula) 1.1.12, the number of

edges is one-half of this sum, which is nr/2.

Examples of Regular Graphs

We now consider some important classes of regular graphs.

1. Complete Graphs, a complete graph is a graph in which each vertex is joined to each of the others by exactly one edge. The complete graph with n vertices is denoted by Kn. The graph Knis regular of degree n− 1, and therefore has n(n − 1)/2 edges, by Theorem 1.1.22.

(30)

Figure 1.10: Complete Graphs.

2. Null Graphs, A null graph is a graph with no edges. The null graph with n vertices is denoted by Nn. The graph Nn is regular of degree 0.

Figure 1.11: Null Graphs.

3. Cycle Graphs, A cycle graph is a graph consisting of a single cycle of vertices and edges. The cycle graph with n vertices is denoted by Cn. The graph Cn is regular of degree 2, and has n edges. For n≥ 3, Cn can be drawn in the form of a regular polygon.

Figure 1.12: Cycle Graphs.

Deﬁnition 1.1.23 If V ={v1, ..., vn} then the degree sequence of G is deg(v1), ..., deg(vn) arranged in decreasing order. For example the degree sequence of P5 is 2, 2, 2, 1, 1.

Deﬁnition 1.1.24 An isomorphism from a simple graph G to a simple graph H is a bijection f : V (G) → V (H) such that uv ∈ E(G) if and only if f(u)f(v) ∈ E(H). We say "G is isomorphic to H", written G ∼= H, if there is an isomorphism from G to H. Deﬁnition 1.1.25 Two graphs are isomorphic to each other, written G ∼= H, if there is a bijection f : VG → VH such that a, b ∈ VG are adjacent if and only if f (a), f (b) ∈ VH are adjacent. For example K3 ∼= C3 and both are called triangles. Also K2 ∼= K1,1 ∼= P2.

(31)

1.1. Graphs 29

Two graphs G and H are isomorphic if H can be obtained by relabelling the vertices of G that is, if there is a one-one correspondence between the vertices of G and those of

H, such that the number of edges joining each pair of vertices in G is equal to the number

of edges joining the corresponding pair of vertices in H. Such a one-one correspondence is an isomorphism.

Example 1.1.26 For example, the graphs G and H represented by the diagrams

Figure 1.13: Two graphs G and H are not the same, but they are isomorphic. are not the same, but they are isomorphic, since we can relabel the vertices in the graph

G to get the graph H, using the following one-one correspondence: G↔ H, u ↔ 4, v ↔ 3, w ↔ 2 and x ↔ 1

Note that edges in G correspond to edges in H, for example: the two edges join-ing u and v in G correspond to the two edges joinjoin-ing 4 and 3 in H; the edge uw in G corresponds to the edge 42 in H; the loop ww in G corresponds to the loop 22 in H. Remark 1.1.27 Note that if G ∼= H then |VG| = |VH|, |EG| = |EH|, and their degree sequences must be identical. However none of these is a suﬃcient condition for isomor-phism.

Example 1.1.28 For example, the graphs G and H represented by the diagrams below (see Figure 1.14); which have|VG| = |VH| and |EG| = |EH|, but they are not isomorphic.

(32)

1.1.2 Planar Graphs

Deﬁnition 1.1.29 A graph is planar if it can be drawn in the plane such that no edges are crossing each other (its edges do not cross). This particular drawing of the graph is called a plane graph. For example K4 is planar graph but K3,3 is not planar graph.

Although the complete graph with four vertices K4 is usually pictured with crossing edges

as in Figure. 1.15(a), it can also be drawn with noncrossing edges as in Figure. 1.15(b); hence K4 is planar.

Figure 1.15: The graph K4 drawn as a plane graph without edge crossing.

Note that if G is disconnected, then G is planar if and only if each component is planar, hence we may assume well that G is connected throughout this thesis.

Proposition 1.1.30 If H ⊆ G and H is not planar then neither is G. In particular Km,n is not planar if m, n≥ 3.

Deﬁnition 1.1.31 A planar graph partitions the plane into subsets called regions (faces). For example the plane graph of K4 has four regions, one of which is exterior to the graph.

1.2 Maps

The aim of this section is to provide a short and accessible presentation of planar maps. For a more detailed introduction, see the introductory chapter in [69] and the thesis of El Marraki [43].

1.2.1 Preliminaries and notations on maps

We begin with some vocabulary on maps. A map is a proper embedding of a connected graph into the two-dimensional sphere, considered as continuous deformations. A map is rooted if one of its edges is distinguished as the root-edge and attributed an orientation. Unless otherwise speciﬁed, all maps under consideration in this thesis are rooted. The face at the right of the root-edge is called the root-face and the other faces are said to be internal. Similarly, the vertices incident to the root-face are said to be external and the others are said to be internal. Graphically, the root-face is usually represented as the inﬁnite face when the map is projected on the plane.

(33)

1.2. Maps 31

Deﬁnition 1.2.1 (Map) A map C is a graph G drawn on a surface X or embedded into it (that is, a compact 2-dimensional orientable variety) in such a way that:

• the vertices of graph are represented as distinct points of the surface.

• the edges are represented as curves on the surface that intersect only at the vertices. • if we cut the surface along the graph thus drawn, what remains (that is, the set X\G)

is a disjoint union of connected components, called faces, each homeomorphic to an open disk (for more information on the faces of a map see [43] and [69]).

A planar map is a map drawn on the plane. Through this thesis, all maps are planar and connected.

Figure 1.16: One graph gives two planar maps

A planar map (hereafter shortly called a map) is an isotopy class of planar embeddings of a connected planar graph. Notice that the graphs embedded are unlabelled. To state it simply, a planar map is a connected unlabelled graph drawn in the plane without edge-crossings and up to continuous deformation. Planar maps are often called plane graphs in the literature [43, 69]. As illustrated in Figure 1.17 (a)-(b), a planar graph can have non-isotopic planar embeddings, so that it gives rise to several maps. Due to the topological embedding, a map has more structure than a graph. In particular, a map has faces, each face corresponding to a connected component of the plane splits by the embedding.

Figure 1.17: (a) A representation of a graph; its set of vertices is{1, 2, 3, 4}, and (multi)set of edges is {{1, 2}, {2, 3}, {2, 4}, {2, 4}, {3, 3}, {3, 4}}. (b) Two embeddings of this graph in the sphere, which are not homeomorphic since the second has a triangular face, unlike the ﬁrst.

(34)

The unique unbounded face is called the outer (or inﬁnite) face. Vertices, edges, and faces are called the 0-cells, 1- cells, and 2-cells of the map, respectively. The numbers|V |,

|E|, and |F | of vertices, edges, and faces (including the outer (external) face) of a map

are related by the well known Euler’s relation:

|V | − |E| + |F | = 2.

1.2.2 Faces and Euler’s formula

LetC be a planar map. If we omit the line segments of C from the plane surface on which

C is drawn, the remainder splits into a number of connected open regions; the closure of

such a region is called a face.

An edge is incident to a face if it belongs to the boundary of this face. If both "banks" of the edge belong to the same face, then such an edge is called an isthmus; we say that an isthmus is incident to the corresponding face twice.

Deﬁnition 1.2.2 The degree of a face f denoted by deg(f ) is the number of edges incident to this face (isthmus being counted twice).

The notion of face degree is illustrated in Figure.1.18). If we go around the boundary of a face slightly inside the face, then the number of times we pass along an edge is exactly the degree of the face.

Figure 1.18: The degree of the faces of this planar map are written inside the faces

The following proposition is similar to the previous proposition 1.1.12.

Proposition 1.2.3 The sum of the degrees of all faces of a map C is equal to twice the number of its edges, i.e., _∑

f∈F_C

deg(f ) = 2|E_C|,

(35)

1.2. Maps 33

Theorem 1.2.4 (Euler characteristic) Let us associate to a map C the number

χ(C) = |V_C| + |F_C| − |E_C|

which is called its Euler characteristic. Then χ(C) does not depend on the map C itself but only on its genus g and is equal to 2− 2g.

1.2.3 Euler’s formula

Euler’s formula for planar maps is:

|VC| + |FC| − |EC| = 2 − 2g,

where F_C is the set of faces of the planar map C and g is the genus of a map C, in the planar case g = 0. A map of genus zero is called planar map. This formula is valid for planar maps (graphs embedded in the plane or in the surface without edge-crossings). For the genus zero case this theorem was already observed by Descartes, and was proved by Euler in 1752 [125].

The following theorem gives another famous result due to Euler.

Theorem 1.2.5 (Euler’s formula) Let G be a connected planar map with n vertices, m edges and f faces (regions). Then n− m + f = 2.

Proof: Let’s apply the induction on m . For m = 0 we have n = 1 and f = 1, so that the statement holds. Now let m ̸= 0. If G contains a cycle, we discard one of the edges contained in this cycle and get a graph G′ with n′ = n, m′ = m− 1 and f′ = f− 1. By induction hypothesis, n′−m′+ f′ = 2 and hence n−m+f = 2. If G is acyclic, then G is a tree so that m = n−1, by Theorem 1.2.8; as f = 1, we still obtain n−m+f = 2.

Example 1.2.6 In the mapC shown in Figure. 1.19, we have: the number of the vertices is 11, the number of the faces is 13, the number of the edges is 22, then 11 + 13− 22 = 2.

(36)

Corollary 1.2.7

1. If G is planar with|V | ≥ 3 then |E| ≤ 3|V | − 6. In particular K5 is not planar and

neither is Kn for all n≥ 6.

2. If G is planar then there is a vertex of degree 5 or less. 3. If G is planar and contains no triangles then |E| ≤ 2|V | − 4.

Remark 1.2.8 In this thesis, we are only interested in planar graphs. For further detail and more explanation on graphs we refer the reader to [1, 10, 13, 19, 20, 26, 39, 56, 117, 118, 119, 121].

1.3 Trees and Forests

In this section, we focus our attention on one particularly important and useful type of graph - a tree. Although trees are relatively simple structures, they form the basis of many of the practical techniques used to model and to design large-scale systems.

The concept of a tree is one of the most important and commonly used ideas in graph theory, especially in the applications of the subject. It arose in connection with the work of Gustav Kirchhoﬀ on electrical networks in the 1840, and later with Arthur Cayley’s work on the enumeration of molecules in the 1870. More recently, trees have proved to be of value in such areas as computer science, decision making, linguistics, and the design of gas pipeline systems.

The word "tree" suggests branching out from a root and never completing a cycle. Trees as graphs have many applications, especially in data storage, searching, and communication. Remark 1.3.1 A path graph Pn is a tree consisting of a single path through all its vertices. The graph Pn has n− 1 edges, and is obtained from the cycle graph Cn by removing any of its edges (see Figure 1.20).

Figure 1.20: Path graphs.

Deﬁnition 1.3.2 (Tree) A tree is a connected simple graph that contains no cycle, i.e., without cycle. For example P4.

Deﬁnition 1.3.3 (A plan tree) A plan tree is a tree designed in the plane or is a map with only one face, the outer face (see Figure 1.21).

(37)

1.3. Trees and Forests 35

Figure 1.21: Simple trees

Remark 1.3.4 Tree graphs form an important class of planar graphs.

In general a graph which contains no cycles is called acyclic. A tree is a connected acyclic graph. A tree is a path if and only if its maximum degree is 2. A star is a tree consisting of one vertex adjacent to all the others K1,n−1, see Figure 1.22.

Figure 1.22: Path and Star trees

An acyclic graph, one not containing any cycles, is called a forest. A connected forest is called a tree (Thus, a forest is a graph whose components are trees). The vertices of degree 1 in a tree are its leaves. Every nontrivial tree has at least two leaves-take, for example, the ends of a longest path. This little fact often comes in handy, especially in induction proofs about trees: if we remove a leaf from a tree, what remains is still a tree. Example 1.3.5 A tree is a connected forest, and every component of a forest is a tree.

1.3.1 Properties of Trees

Trees have many equivalent characterizations, any of which could be taken as the deﬁnition. Such characterizations are useful because we need only verify that a graph satisﬁes anyone of them to prove that it is a tree, after which we can use all the other properties.

We ﬁrst prove that deleting a leaf from a tree yields a smaller tree.

Proposition 1.3.6 A tree contains a vertex of degree 1, which is called a leaf.

Lemma 1.3.7 Every tree with at least two vertices has at least two leaves. Deleting a leaf from an n-vertex tree produces a tree with n− 1 vertices.

(38)

Proof: A connected graph with at least two vertices has an edge. In an acyclic graph, an endpoint of a maximal nontrivial path has no neighbor other than its neighbor on the path. Hence the endpoints of such a path are leaves. Let v be a leaf of a tree G, and let G′ = G− v. A vertex of degree 1 belongs to no path connecting two other vertices. Therefore, for u, w ∈ V (G′), every u, w-path in G is also in G′. Hence G′ is connected. Since deleting a vertex cannot create a cycle, G′ also is acyclic. Thus G′ is a tree with

n−1 vertices.

Remark 1.3.8 The previous Lemma implies that every tree with more than one vertex arises from a smaller tree by adding a vertex of degree 1 (all our graphs are ﬁnite). This rescues some proofs from the induction trap: growing an n + 1-vertex tree from an arbitrary n-vertex tree by adding a new neighbor at an arbitrary old vertex generates all trees with n + 1 vertices. The word "arbitrary" means that the discussion considers all ways of making the choice.

The proof of equivalence of characterizations of trees uses induction, prior results, a counting argument, extremality, and contradiction.

Theorem 1.3.9 For an n-vertex graph G (with n≥ 1), the following are equivalent (and characterize the trees with n vertices).

(A) G is connected and has no cycles (G is acyclic). (B) G is connected and has n− 1 edges (|V | = |E| + 1). (C) G has n− 1 edges and no cycles.

(D) For u, v∈ V (G), G has exactly one u, v-path.

Proof: We ﬁrst demonstrate the equivalence of A, B, and C by proving that any two of connected, acyclic, n− 1 edges together imply the third. A ⇒ B, C. We use induction on n. For n = 1, an acyclic 1-vertex graph has no edge. For n > 1, we suppose that the implication holds for graphs with fewer than n vertices. Given an acyclic connected graph

G, the previous Lemma provides a leaf v and states that G′ = G− v also is acyclic and connected (see ﬁgure above). Applying the induction hypothesis to G′ yields EG′ = n−2. Since only one edge is incident to v, we have EG = n− 1.

B⇒ A, C. Delete edges from cycles of G one by one until the resulting graph G′is acyclic. Since no edge of a cycle is a cut-edge (the previous Theorem), G′ is connected. Now the

(39)

1.3. Trees and Forests 37

preceding paragraph implies that EG′ = n− 1. Since we are given EG = n− 1, no edges were deleted. Thus G′ = G, and G is acyclic.

C ⇒ A, B. Let Gl, ..., Gk be the components of G. Since every vertex appears in one component, ∑

i

n(Gi) = n. Since G has no cycles, each component satisﬁes property A. Thus EGi = n(Gi)− 1. Summing over i yields EG=

∑ i

[n(Gi)− 1] = n − k. We are given

EG = n− 1, so k = 1, and G is connected.

A ⇒ D. Since G is connected, each pair of vertices is connected by a path. If some pair is connected by more than one, we choose a shortest (total length) pair P , Q of distinct paths with the same endpoints. By this extremal choice, no internal vertex of P or Q can belong to the other path (see ﬁgure below).

This implies that P ∪ Q is a cycle, which contradicts the hypothesis A.

D ⇒ A. If there is a u, v-path for every u, v ∈ VG, then G is connected. If G has a cycle

C, then G has two u, v-paths for u, v ∈ VC; hence G is acyclic (this also forbids loops). Deﬁnition 1.3.10 A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. We write G− e or G − M for the subgraph of G obtained by deleting an edge e or set of edges M . We write G− v or G − S for the subgraph obtained by deleting a vertex v or set of vertices S.

Next, we characterize cut-edges in terms of cycles.

Theorem 1.3.11 An edge is a cut-edge if and only if it belongs to no cycle. Corollary 1.3.12 a) Every edge of a tree is a cut-edge.

b) Adding one edge to a tree forms exactly one cycle. c) Every connected graph contains a spanning tree.

Proof: (a) A tree has no cycles, so Theorem 1.3.11 implies that every edge is a cut-edge. (b) A tree has a unique path linking each pair of vertices (Theorem 1.3.9 D), so joining two vertices by an edge creates exactly one cycle. (c) As in the proof of B

⇒ A, C in Theorem 1.3.9, iteratively deleting edges from cycles in a connected graph

(40)

Theorem 1.3.13 The following assertions are equivalent for a graph T : (i) T is a tree;

(ii) any two vertices of T are linked by a unique path in T ;

(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge e∈ T ;

(iv) T is maximally acyclic, i.e. T contains no cycle but T + xy does, for any two non-adjacent vertices x, y ∈ T .

1.3.2 Spanning Trees

An important concept that we need later is that of a spanning tree in a graph.

Deﬁnition 1.3.14 Let G be a connected graph. Then a spanning tree in G is a subgraph of G that includes every vertex and is also a tree. For example, the following diagram shows a graph G and its three spanning trees.

Figure 1.23: A graph G with its 3 spanning trees

1.4 Distance in trees and graphs

When using graphs to model communication networks, we want vertices to be close to-gether to avoid communication delays. We measure distance using lengths of paths. Many applications of graphs involve getting from one vertex to another. For example, we may wish to find the shortest route between one town and another. Other examples include the routing of a telephone call between one subscriber and another, the flow of current between two terminals of an electrical network, and the tracing of a maze. We now make this idea precise by defining a walk in a graph. It is sometimes useful to be able to refer to a walk under more restrictive conditions in which we require all the edges, or all the vertices, to be different.

(41)

1.4. Distance in trees and graphs 39

Deﬁnition 1.4.1 (Walk) A walk is a sequence of edges v1v2, v2v3, ..., vn−1vn which are not necessarily distinct (unless the walk is a path). In this case we say that the walk is from v1 to vn of length n− 1.

Deﬁnition 1.4.2 (Distance) The distance between two distinct vertices vi and vj of a graph G, denoted by d(vi, vj) is equal to the length of the shortest path (number of edges in) that connects vi and vj (the least length between vi and vj). Conventionally,

d(vi, vi) = 0. If G has a u, v-path, then the distance from u to v, written dG(u, v) or sim-ply d(u, v), is the least length of a u, v-path. If G has no such path, then d(u, v) = ∞ [87]. The distance between two vertices v and u, written d(v, u), is the length of the shortest walk from v to u, if it exists, otherwise let d(v, u) =∞. Note that the shortest walk is necessarily a path. Furthermore, in a weighted graph, d(v, u) is understood to be the minimum total weights of all possible walks from v to u.

Deﬁnition 1.4.3 (Weight) The weight, denoted by p(vi, vj) is the number of edges that connects vi with vj.

We use the word "diameter" due to its use in geometry, where it is the greatest distance between two elements of a set.

Deﬁnition 1.4.4 (Diameter) The diameter of a graph G is deﬁned as the maximum of the set of all shortest walks joining any two vertices, i.e.,

Diam(G) = max{d(v, u)|v, u ∈ V }.

For example diam(G) = 1 if and only if G is complete but the diameter in a plane tree is the number of edges of the longest path. Note also that diam(G) = ∞ if and only if G is disconnected.

Example 1.4.5 The diameter of the graph below is 2 (see Figure. 1.24). This is because for any vertex not directly connected to another, there is a path of length two connecting the two. By looking at the graph, it can be seen that this is true.

(42)

(43)

Chapter 2 Spanning trees

2.1 Introduction

Spanning trees have always been of great interest in various areas of computer science. The same is true for the idea of shortest paths in a graph. The number of spanning trees of a graph (network) G, also known as the complexity, denoted by τ (G), is an important, well-studied quantity in graph theory, and appears in a number of applications. Most notable application fields are network reliability. In order to motivate the research in this thesis, we begin with some basic definitions and properties about spanning tree. An important concept which is needed in this chapter is the spanning tree in a graph. For further detail and more explanation on properties of trees and graphs we refer the reader to [117]. In this chapter, we are going to begin by introducing the basic notions and definitions needed for spanning trees and to study many different properties of them.

2.2 Basic concepts and research background

The central concept of the research presented here the spanning tree. A spanning tree in a graph G is a spanning subgraph of G, which is a tree. We deﬁne a spanning tree formally as follows.

Deﬁnition 2.2.1 For a graph G, a spanning tree in G is a tree which has the same vertex set as G (tree that passing through all the vertices of the graph), i.e., a spanning tree T of a graph G = (V, E) is a graph T = (V, E′) such that T is a tree and E′ ⊆ E. (see Figure. 2.1).

Remark 2.2.2 A spanning tree of a graph G is a connected subgraph without cycles that includes every vertex of G, assuming G is connected. Otherwise, G has spanning forests, which are maximal subgraphs without cycles (so a forest is a subgraph without cycles). Here we shall discuss spanning trees since we are only concerned with connected graphs. Figure 2.1 gives an example of spanning trees of a graph. The subgraphs on the right is spanning trees of the graph on the left.

(44)

Figure 2.1: A graph(left) and all spanning trees (right) of this graph.

The number of spanning trees of a ﬁnite graph or multigraph G, also known as the complexity τ (G), is certainly one of the most important graph-theoretical parameters, and also one of the oldest. Its applications range from the theory of networks, where the number of spanning trees is used as a measure for network reliability [34] to theoretical chemistry, in connection with the enumeration of certain chemical isomers [22]. Kirchhof’s celebrated matrix tree theorem [65] that relates the properties of an electrical network to the number of spanning trees in the underlying graph.

The number of spanning trees in a graph (network) is an important, well studied quan-tity [36]. As well as being of combinatorial interest, several application uses, mentioned in the following, are adduced in [62].

1. Kirchhoﬀ’s laws, well know as Matrix Tree Theorem, provide an eﬀective method for designing electrical circuits, which are enormously useful in the analysis and synthesis of networks.

2. Suppose we are given a network of communication lines, which can break. The probability of a single line breaking is 1−p. It is necessary to estimate the reliability of such a network. If the reliability is the probability of connectedness of the network,

P , then P = m ∑ k=n−1 Akpk(1− p)m−k,

where n is the number of vertices, m the number of edges, of the graph; Ak is the number of connected subgraphs with n vertices and k edges. It is clear that, if the reliability of each line is small, then

P ≈ An−1pn−1(1− p)m−n+1, where An₋₁ is the number of spanning trees of the graph.

(45)

2.2. Basic concepts and research background 43

Thus, with low reliability of each of the line, the network’s reliability is de-termined, basically, by the number of spanning trees of the network.

3. In building a maser, one must investigate the possible particle transitions. For this, one constructs a graph in which the vertices correspond to energy levels and edges to possible particle transitions. Then for the analysis of the maser’s energetics, it turns out to be very useful to know the number of spanning trees in the corresponding graph.

The research of the number of spanning trees in a graph has a long history. The cornerstone of the research, the Matrix Tree Theorem, dated back to 1847, is attributed to Kirchhoﬀ. Most research about the number of spanning trees is devoted to determining exact formulae for the number of spanning trees in many kinds of special graphs, see [6, 14, 62, 63, 65, 66, 127]. We now state the general methods for counting the number of spanning trees in graphs in the following.

Now, the problem is: given a map (graph embedded into surfaces), how many spanning trees does it have? As example, a graph G with all its spanning trees is displayed in Figure 2.2.

Figure 2.2: A graph G gives rise to ﬁve spanning trees

Let G be a connected graph. Then a spanning tree in G is a subgraph of G that includes every vertex and is also a tree. For example, the following diagram shows a graph G and three of its spanning trees; (see Figure. 2.3).

(46)

The number of spanning trees in a graph can be very large; for example, the above graph G has twenty-one spanning trees and the Petersen graph has 2000 labelled spanning trees. Given a connected graph, we can construct a spanning tree by using either of the following two methods. We illustrate these by applying them to the graph G above. Building-up method:

Select edges of the graph one at a time, in such a way that no cycles are created; repeat this procedure until all vertices are included.

Example 2.2.3 In the above graph G in Figure 2.3, we select the edges vz, wx, xy, yz; then no cycles are created. We obtain the following spanning tree; (see Figure. 2.4).

Figure 2.4: A spanning tree of graph G

Cutting-down method:

Choose any cycle and remove any one of its edges; repeat this procedure until no cycles remain.

Example 2.2.4 From the above graph G in Figure 2.3, we remove the edges vy (destroy-ing the cycle vwyv), yz (destroy(destroy-ing the cycle vwyzv), xy (destroy(destroy-ing the cycle wxyw). We obtain the following spanning tree; (see Figure. 2.5).

(47)

2.3. Spanning Trees and Enumeration 45

Remark 2.2.5 If two spanning trees in a graph have diﬀerent edge (arc) sets, we consider them as diﬀerent trees, even though, they may be isomorphic.

2.3 Spanning Trees and Enumeration

There are 2(n2) simple graphs with vertex set [n] = {1, ..., n}, since each pair may or may

not form an edge. How many of these are trees? In this section, we solve this counting problem, count spanning trees in arbitrary graphs, and discuss several applications.

2.3.1 Enumeration of Trees

With one or two vertices, only one tree can be formed. With three vertices there is still only one isomorphism class, but the adjacency matrix is determined by which vertex is the center. Thus there are three trees with vertex set [3]. With vertex set [4], there are four stars and 12 paths, yielding 16 trees. With vertex set [5], a careful study yields 125 trees.

Now we may see a pattern. With vertex set [n], there are nn−2 trees; this is Cay-ley’s Formula. Prüfer, Kirchhoﬀ, Pólya, Renyi, and others found proofs of that while J.W. Moon [92] wrote a book about enumerating classes of trees. We present a bijective proof, establishing a one-to-one correspondence between the set of trees with vertex set [n] and a set of known size.

Remark 2.3.1 In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edges or vertices, or both of a graph.

Theorem 2.3.2 (Cayley’s Formula [1889]). For a set S ⊆ N of size n, there are nn−2 trees with vertex set S.

Corollary 2.3.3 The number of labeled trees with n ≥ 2 vertices is the number of spanning trees of a labeled Kn, which is nn−2.

2.3.2 Spanning trees in graphs

We can interpret Cayley’s Formula in another way. Since the complete graph with vertex set [n] has all edges that can be used in forming trees with vertex set [n], the number of trees with a speciﬁed vertex set of size n equals the number of spanning trees in a complete graph on n vertices.

We now consider the more general problem of computing the number of spanning trees in any graph G. In general, G will not have as much symmetry as a complete graph, so it is not reasonable to expect as simple a formula as for Kn, but we can hope for an algorithm that provides a simple way to compute the answer for a given graph G.

(48)

Example 2.3.4 Below is the kite. To count the spanning trees, observe that four are paths around the outside cycle in the drawing. The remaining spanning trees use the diagonal edge. Since we must include an edge to each vertex of degree 2, we obtain four more spanning trees. The total is eight.

Figure 2.6: A graph G and its all spanning trees

In Example 2.3.4, we counted separately the trees that did or did not contain the diagonal edge. This suggests a recursive procedure to count spanning trees. It is clear that the spanning trees of G not containing e are simply the spanning trees of G− e, but how do we count the trees that contain e? The answer uses an elementary operation on graphs. Remark 2.3.5 We denote by τ (G) the number of spanning trees of a graph G, sometimes called the complexity of G. If G is not connected then τ (G) = 0, so we can assume that

G is connected from now on. If G′ is obtained from G by removing all the loops of G, then τ (G′) = τ (G), since a loop can never occur in a spanning tree, i.e., loops don’t aﬀect spanning trees, so we delete them before the computation. Thus, we may assume that G contains no loops as well. Multiple edges, however, do remain a possibility.

2.4 Matrices associated to a graph

How do we specify a graph? We can list the vertices and edges (with end-points), but there are other useful representations. Saying that a graph is loopless means that multiple edges are allowed but loops are not. We need to deﬁne some matrices as follows.

2.4.1 Adjacency Matrix

The adjacency matrix of a graph G is the square matrix A = A(G) indexed by V × V , which has its entries as: Avv = 0 for all v∈ V , and if v ̸= u in V then Avu is the number of edges of G which have vertices v and u at their ends. In other words, the adjacency matrix A of a graph G, is a (0, 1)-matrix, where 1’s represent adjacent vertices. A has a row for each vertex, and a column for each vertex. If a vertex v is connected to a vertex

u, the entry in row v, column u is 1, and so is the entry in row u, column v. For instance,

Enumaration of the number of spanning trees in some special planar maps

ENUMERATION OF THE NUMBER OF SPANNING TREES

IN SOME SPECIAL PLANAR MAPS

ACKNOWLEDGEMENTS

Contents

I

Preliminaries

19

II

Theoretical Part

53

III

Use of Derived Theoretical Results

83

List of Figures

List of Tables

Introduction

Part I

Preliminaries

Chapter 1

Deﬁnitions and Properties

1.1

Graphs

1.1.1

Vertex Degrees

1.1.2

Planar Graphs

1.2

Maps

1.2.1

Preliminaries and notations on maps

1.2.2

Faces and Euler’s formula

1.2.3

Euler’s formula

1.3

Trees and Forests

1.3.1

Properties of Trees

1.3.2

Spanning Trees

1.4

Distance in trees and graphs

Chapter 2

Spanning trees

2.1

Introduction

2.2

Basic concepts and research background

2.3

Spanning Trees and Enumeration

2.3.1

Enumeration of Trees

2.3.2

Spanning trees in graphs

2.4

Matrices associated to a graph

2.4.1

Adjacency Matrix