3. Graft transformations that decrease the distance spectral radius

https://doi.org/10.1051/ro/2021085 www.rairo-ro.org

ON GRAFT TRANSFORMATIONS DECREASING DISTANCE SPECTRAL RADIUS OF GRAPHS

Yanna Wang

and Bo Zhou

Abstract.The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. In this paper, we give several less restricted graft transformations that decrease the distance spectral radius, and determine the unique graph with minimum distance spectral radius among home- omorphically irreducible unicylic graphs onn≥6 vertices, and the unique tree with minimum distance spectral radius among trees onnvertices with given number of vertices of degree two, respectively.

Mathematics Subject Classification. 05C50, 15A48.

Received November 21, 2020. Accepted May 27, 2021.

1. Introduction

We consider simple, finite, undirected and connected graphs. Foru∈V(G), letNG(u) be the set of neighbors ofuinG. The degree of a vertexuin G, denoted by deg_G(u), is the number of edges incident touinG.

Ahomeomorphically irreducible treeis a tree with no vertex of degree two [6]. Ahomeomorphically irreducible unicylic graphis a unicylic graph with no vertex of degree two.

Let G be a connected graph on n vertices. For u, v ∈ V(G), the distance between u and v in G, denoted by dG(u, v), is the length of a shortest path connecting them in G. In particular, dG(u, u) = 0. The distance spectrum of G is the spectrum of thedistance matrix of G, defined as the nby n symmetric matrix D(G) = (d_G(u, v))_u,v∈V(G). The distance spectral radius of G, denoted by ρ(G), is the largest distance eigenvalue of G. Graham and Pollack [5] studied the distance spectrum for the first time due to its connection to a data communication problem. Now the distance spectrum has been studied extensively, see the survey [1]. Particularly, the distance spectral radius has received much attention. Graphs with minimum and/or maximum distance spectral radius have been determined for some classes of graphs, see,e.g., [2,7,11–15]. Yuet al.[16] determined the graphs with minimum and maximum distance spectral radius among unicylic graphs. Lin and Zhou [8]

determined the trees with maximum distance spectral radius among trees on nvertices with given number of vertices of degree two.

To determine the structure of the graphs with minimum or maximum distance spectral radius in some class of graphs, we usually suppose the graph does not have the structure and perform surgery to obtain a graph for

Keywords.Distance spectral radius, distance matrix, graft transformation, homeomorphically irreducible unicylic graph.

1 Basic Courses Department, Guangdong Communication Polytechnic, Guangzhou 510650, P.R. China.

2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China.

∗Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, ROADEF, SMAI 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

which the distance spectral radius is decreased or increased. Such surgery is known as graft transformation(s) in the literature, see,e.g., [3,4,9,14,16].

In this paper, we propose some graft transformations in a more elaborate way. We propose some graft transformations with less restricted conditions that decrease the distance spectral radius, and as applications, we identify the unique graphs that minimize the distance spectral radius among homeomorphically irreducible unicylic graphs onn≥6 vertices, and among trees on nvertices with given number of vertices of degree two, respectively.

2. Preliminaries

Let G be a connected graph with V(G) = {v1, . . . , vn}. Since D(G) is irreducible, by Perron–Frobenius theorem, ρ(G) is simple and there is a unique unit positive eigenvector corresponding to ρ(G), which is called the distance Perron vector of G, denoted by x(G). If y = (yv₁, . . . , yv_n)^> ∈ Rⁿ is unit and has at least one nonnegative entry, then by Rayleigh’s principle, we haveρ(G)≥y^TD(G)ywith equality if and only ify=x(G).

If x = x(G), then for each u ∈ V(G), we have ρ(G)xu = P

v∈V(G)dG(u, v)xv, which is called the distance eigenequationofGatu.

LetN ⊆(NG(u)\NG(v))\ {v}. Let G⁰ =G−uw+vw for w∈N. We say that G⁰ is obtained fromGby moving edgeuw atufromutov.

For a connected graphGwithV₁⊆V(G), letσ_G(V₁) be the sum of the entries of the distance Perron vector ofGcorresponding to the vertices inV1. Furthermore, if all the vertices ofV1induce a connected subgraphH ofG, then we writeσG(H) instead of σG(V1).

Acomponentof a graph is a maximal connected subgraph, and acut edgeis an edge of a graph whose removal increases the number of components of the graph.

For a connected graphG, lets(G) be the minimum row sum ofD(G). In [17], ^s(G)_n is called the mean vertex deviation ofG, wheren=|V(G)|. It is known thatρ(G)≥s(G), see Theorem 1.1 in page 24 of [10].

3. Graft transformations that decrease the distance spectral radius

Firstly, we give a result related to (the entries of) the distance Perron vector, which will be frequently used in the subsequent proofs.

Lemma 3.1. Suppose thatv, wbe two non-adjacent neighbors of vertexuin a connected graphG. Letx=x(G).

Thenx_w+x_u−x_v>0.

Proof. LetV1=V(G)\ {u, v, w}. Forz∈V1, one hasdG(w, z)≥1 anddG(u, z)−dG(v, z)≥ −dG(u, v) =−1, sodG(w, z) +dG(u, z)−dG(v, z)≥0. From the distance eigenequations ofGatw, uandv, we have

ρ(G)x_w=x_u+ 2x_v+ X

z∈V1

d_G(w, z)x_z, ρ(G)xu=xw+xv+X

z∈V1

dG(u, z)xz, ρ(G)x_v= 2x_w+x_u+X

z∈V1

d_G(v, z)x_z. Thus

ρ(G)(xw+xu−xv) =−xw+ 3xv+ X

z∈V1

(dG(w, z) +dG(u, z)−dG(v, z))xz

≥ −xw+ 3xv,

which implies (ρ(G) + 1)(x_w+x_u−x_v)≥x_u+ 2x_v>0. So it follows thatx_w+x_u−x_v >0.

Now we turn our attention to some graft transformations that decrease the distance spectral radius.

Theorem 3.2. Let Gbe a graph consisting of nontrivial connected graphs G₁ andG₂ sharing a unique vertex usuch thatE(G) =E(G1)∪E(G2). Suppose thatuhas neighbor v of degree at least two inG2 satisfying that, for anyz∈V(G2)\ {u, v},dG(u, z)6=dG(v, z). Let G⁰ be the graph obtained fromGby moving all the edges at v except uv fromv tou. Thenρ(G)> ρ(G⁰).

Proof. Letwbe a neighbor ofuinG₁. Letx=x(G⁰). By Lemma3.1, we havex_w+x_u−x_v>0.

LetS ={z∈V(G₂)\ {u, v}:d_G(u, z)−d_G(v, z) = 1}. Let z be a neighbor ofv in G₂ different fromu. As dG(u, z)6=dG(v, z) = 1 anddG(u, z)≤dG(v, z) +dG(u, v) = 2, one hasdG(u, z) = 2. Soz∈S and S6=∅.

Claim. ¹₂x^>(D(G)−D(G⁰))x≥σ_G⁰(S) (σ_G⁰(G₁)−x_v).

Note first that, as we pass fromGtoG⁰, the distance between a vertex ofSand a vertex ofV(G1) is decreased by 1, and the distance between a vertex of S andv is increased by 1. So, to prove the claim, we need only to show that the distance between any other vertex pairs is decreased or remains unchanged.

It is evident that the distance between any two vertices inV(G₁)∪ {v}remains unchanged as we pass from GtoG⁰.

Suppose that z1, z2 ∈V(G2)\ {u, v}. Let P be a path from z1 to z2 with length dG(z1, z2) in G. If v lies outsideP, thenP is also a path connectingz1 andz2 inG⁰. Suppose thatv lies onP. Ifulies outsideP, then the path obtained from P by replacingv withuis a path connectingz₁ and z₂ inG⁰. Otherwise, ulies onP. In this case,uvappears to be an edge onP. So the path obtained fromP by deletingvis a path connectingz₁ andz2 inG⁰. So the distance between any two vertices inV(G2)\ {u, v}is decreased or remains unchanged as we pass fromGtoG⁰.

Suppose thatz∈S:= (V(G2)\{u, v})\S(ifS6=∅). ThendG(u, z)−dG(v, z)6= 0,1. As|dG(u, z)−dG(v, z)| ≤ d_G(u, v) = 1, one hasd_G(u, z)−d_G(v, z) =−1. LetP be a path fromutoz with lengthd_G(u, z) inG. Thenv lies outsideP, soP is also a path fromutozinG⁰. Therefore, the distance between a vertex inS and a vertex inV(G1)∪ {v} is decreased or remains unchanged as we pass fromGtoG⁰.

Now we complete the proof of the claim. Asρ(G)≥x^>D(G)xandρ(G⁰) =x^>D(G⁰)x, one hasρ(G)−ρ(G⁰)≥ x^>(D(G)−D(G⁰))x. So, by the claim,

1

2(ρ(G)−ρ(G⁰))≥1

2x^>(D(G)−D(G⁰))x

≥σG⁰(S) (σG⁰(G1)−xv)

≥σ_G⁰(S) (x_w+x_u−x_v)

>0,

and thusρ(G)> ρ(G⁰).

In the following, we give some consequences of Theorem3.2.

Corollary 3.3([14]). LetGbe a connected graph with a cut edge uvthat is not a pendant edge. Let G⁰ be the graph obtained fromGby moving all edges at v except uv fromv tou. Thenρ(G)> ρ(G⁰).

Proof. LetG1be the component ofG−uvcontaininguandG2be the subgraph ofGinduced byV(G)\(V(G1)\

{u}). Note thatdG(u, z) =dG(v, z) + 1 for anyz∈V(G2)\ {u, v}. So the result follows from Theorem3.2.

A chain in a graphGis a cycleC such that G−E(C) has exactly |V(C)|components. Length of the cycle C is the length of the chain. The following is Lemma 3.3 of [3].

Corollary 3.4. Let Gbe a connected graph with a chainC of even length. Let uvbe an edge on the chain C.

Suppose thatdeg_G(u)≥3. IfG⁰=G− {vw:w∈NG(v)\ {u}}+{uw:w∈NG(v)\ {u}}, thenρ(G)> ρ(G⁰).

Proof. LetG₁ be the component ofG−E(C) containingu, andG₂ be the subgraph ofGinduced byV(G)\ (V(G1)\ {u}). Let z∈V(G2)\ {u, v}. Note thatdG(u, z)−dG(v, z) =dG(u, w)−dG(v, w) for somew onC.

Ifz lies onC, this is evident as w=z, otherwise,w is the vertex onCsuch that its distance toz is minimum among all the distances between vertices of C and z in G. As C is a chain, the shortest path connecting u (v, respectively) and w contains only vertices on C. As the length of C is even, there is a shortest path connectinguandwpassing throughv or a shortest path connectingv andwpassing throughu, implying that

|dG(u, w)−dG(v, w)|= 1, sodG(u, z)6=dG(v, z). So the result follows from Theorem3.2.

Corollary 3.5. Let H be a graph consisting of two nontrivial connected graphs H₁ and H₂ sharing a unique vertex usuch that E(H) =E(H₁)∪E(H₂). Suppose that uv₁, . . . , uv_k are pendant edges in H₁, where k ≥1 and that NH₂(u) =N1∪N2, whereN1, N26=∅ andN1∩N2=∅. Let

G=H− {uw:w∈N1}+{vkw:w∈N1} or

G=H− {uw:w∈N2}+{vkw:w∈N2}.

For any vertexw∈V(H₂)\ {u}, if all the paths fromutowwith the lengthd_H(u, w)pass only through vertices inN1 or pass only through vertices inN2, thenρ(G)> ρ(H).

Proof. Assume that G = H− {uw : w ∈ N2}+{vkw : w ∈ N2}. Then H is obtainable from G by moving all the edges at v_k except uv_k from v_k to u. LetG₁ =H₁−v_k and let G₂ be the subgraph of G induced by V(H₂)∪ {vk}. Suppose that z ∈ V(G₂)\ {u, vk}, i.e., z ∈ V(H₂)\ {u}. Note that any shortest path from u to z in H2 goes through only vertices in N1 or N2. Correspondingly, any shortest path fromu to z in G2

goes through only vertices in N1, so dG(u, z) = dG(vk, z)−1 < dG(vk, z), or any shortest path from u to z in G2 goes through only vertices in N2, so dG(u, z) = dG(vk, z) + 1 > dG(vk, z). Now by Theorem 3.2,

ρ(G)> ρ(H).

Ifk≥2, then Corollary3.5becomes Theorem 2.4 of [16].

Corollary3.3may be generalized as the following version.

Theorem 3.6. Let Gbe the graph obtained from vertex disjoint nontrivial connected graphs G1 andG2 with u∈V(G₁)andv∈V(G₂)by adding a pathP_t=v₁. . . v_twithv₁=uandv_t=v, wheret≥2,V(G₁)∩V(P_t) = {v1} andV(G₂)∩V(P_t) ={vt}. LetG⁰ be the graph obtained fromG by moving all the edges at v_t in E(G₂) fromvttov1. Thenρ(G)> ρ(G⁰).

Proof. Letx=x(G⁰). Letp=b₂^tcandp₁=d₂^te.

Let Γ =Pp

i=1xv_i+σG⁰(V(G1)\ {v1}) +σG⁰(V(G2)\ {vt})−Pt

i=p₁+1xv_i. From the distance eigenequations ofG⁰ at vp₁+1 andvp, we have

ρ(G⁰) xv_{p1 +1}−xv_p

= (p1+ 1−p)Γ. (3.1)

Fori= 1, . . . , p−1, from the distance eigenequations ofG⁰ at v_t+1−i,vi,v_t+1−(i+1) andvi+1, we have ρ(G⁰) xvt+1−i−xv_i

− xv_t+1−(i+1)−xv_i+1

=ρ(G⁰) xvt+1−i−xv_i

−ρ(G⁰) xv_t+1−(i+1)−xv_i+1

= 2





i

X

j=1

x_vj+σ_G0(V(G₁)\ {v1}) +σ_G0(V(G₂)\ {vt})−

t

X

j=t+1−i

x_vj





= 2Γ−2

p

X

j=i+1

xv_j + 2

t−i

X

j=p₁+1

xv_j

= 2Γ + 2

p

X

j=i+1

xvt+1−j −xv_j

. (3.2)

We claim that xv_t+1−i−xv_i and Γ have common sign fori = 1, . . . , p by induction on i. If i =p, then it follows from (3.1). Suppose that 1≤i≤p−1, andx_vt+1−j−x_vj and Γ have common sign fori+ 1≤j≤p. So Pp

j=i+1(xvt+1−j−xvj) and Γ have common sign. Thus, from (3.2), (xvt+1−i−xvi)−(xv_t+1−(i+1)−xvi+1) and Γ have common sign. This, together with the induction assumption thatxv_t+1−(i+1)−xv_i+1 and Γ have common sign, implies thatxvt+1−i−xv_i and Γ have common sign.

Note that

Γ>

p

X

i=1

x_vi−

t

X

i=p₁+1

x_vi =−

p

X

i=1

x_vt+1−i−x_vi .

This requires the above common sign to be +. Again, from (3.1) and (3.2), we havexvt+1−i−xv_i> xv_t+1−(i+1)− xv_i+1 > 0 for i = 1, . . . , p−1, i.e., 0 < xvt+1−i −xv_i < xvt+2−i −xvi−1 for i = 2, . . . , p. It follows that for i= 2, . . . , p,

0< xvt+1−i−xvi< xvt−xv1. (3.3) As Pt=v1. . . vtis a proper induced subgraph of G⁰ and dP_t(vi, vj) =dG⁰(vi, vj) for any 1≤i < j≤t, we haves(G⁰)> s(Pt). Note thatρ(G⁰)≥s(G⁰) ([10], Thm. 1.1 in p. 24) ands(Pt) =j

t² 4

k

[17]. So ρ(G⁰)≥s(G⁰)> s(Pt) =

t² 4

· (3.4)

As we pass from Gto G⁰, the distance between a vertex of V(G2)\ {vt} and a vertex of V(G1)\ {v1} is decreased byt−1, the distance between a vertex ofV(G2)\ {vt}andvifori= 1, . . . , tis decreased byt−2i+ 1, and the distances between all other vertex pairs remain unchanged. LetA= (t−1)σ_G⁰(V(G₁)\{v₁}) +Pp

i=1(t− 2i+ 1)(x_vi−x_vt+1−i),i.e.,

A= (t−1)σG⁰(V(G1)\ {v1}) +

t

X

i=1

(t−2i+ 1)xv_i. So

1

2(ρ(G)−ρ(G⁰))≥ 1

2x^>(D(G)−D(G⁰))x

=σ_G⁰(V(G₂)\ {v_t}) (t−1)σ_G⁰(V(G₁)\ {v₁}) +

t

X

i=1

(t−2i+ 1)x_vi

!

=σG⁰(V(G2)\ {vt})·A. (3.5)

LetG^∗be the graph obtained from Gby moving all the edges atv₁ inE(G₁) fromv₁to v_t. Lety=x(G^∗).

LetB= (t−1)σG^∗(V(G2)\ {vt}) +Pp

i=1(t−2i+ 1)(yvt+1−i−yv_i),i.e., B= (t−1)σG^∗(V(G2)\ {vt})−

t

X

i=1

(t−2i+ 1)yv_i. By the similar arguments as above, we have

1

2(ρ(G)−ρ(G^∗))≥ 1

2y^>(D(G)−D(G^∗))y

=σG^∗(V(G1)\ {v1}) (t−1)σG^∗(V(G2)\ {vt})−

t

X

i=1

(t−2i+ 1)yv_i

!

=σG⁰(V(G1)\ {v1})·B. (3.6)

It is evident thatφ:V(G⁰)→V(G^∗) defined by φ(w) =

(w if w∈V(G1)\ {u}or w∈V(G2)\ {v}, vt+1−i if w=vi withi= 1, . . . , t

is an isomorphism fromG⁰ toG^∗. Let M be the permutation matrix associated to this isomorphism. That is, the nonzero entries ofM areMww= 1 ifw∈V(G1)\ {u}or w∈V(G2)\ {v},Mv_iv_t+1−i = 1 for i= 1, . . . , t.

ThenM^TD(G⁰)M =D(G^∗). As ρ(G⁰) =x^>D(G⁰)x= (M x)^>D(G^∗)M x,M xis the distance Perron vector of G^∗and soy=M xby the Perron–Frobenius theorem. That is,y_w=x_wifw∈V(G₁)\ {u}orw∈V(G₂)\ {v}, and yv_i = xvt+1−i. Then B = (t−1)σG⁰(V(G2)\ {vt}) +Pp

i=1(t−2i+ 1)(xv_i−xvt+1−i). From the distance eigenequations ofG⁰ atvtandv1, we have

ρ(G⁰)(x_vt−x_v1) = (t−1) (σ_G⁰(V(G₁)\ {v₁}) +σ_G⁰(V(G₂)\ {v_t})) +

p

X

i=1

(t−2i+ 1) xv_i−xvt+1−i

=A+B−

p

X

i=1

(t−2i+ 1) xv_i−xv_t+1−i

=A+B+

p

X

i=1

(t−2i+ 1) xvt+1−i−xv_i

.

Then, by (3.3) and (3.4), we have

A+B=ρ(G⁰) (x_vt−x_v1)−

p

X

i=1

(t−2i+ 1) x_vt+1−i−x_vi

≥ρ(G⁰) (xv_t−xv₁)−

p

X

i=1

(t−2i+ 1) (xv_t −xv₁)

= ρ(G⁰)−

p

X

i=1

(t−2i+ 1)

!

(xv_t−xv₁)

=

ρ(G⁰)− t²

4

(x_vt−x_v1)

>0.

ThusA >0 orB >0. Now the result follows from (3.5) and (3.6).

Now we present the third graft transformation and consider its effect on the distance spectral radius.

Theorem 3.7. Let G be a graph consisting of two nontrivial connected graphs G1 and G2 sharing a unique vertexusuch that E(G) =E(G₁)∪E(G₂). Suppose thatN_G2(u) ={v₁, v₂},deg_G

2(v_i)≥2 fori= 1,2,v₁ and v₂ are not adjacent, and for any w∈V(G₂)\ {u},d_G2(v₁, w)6=d_G2(v₂, w). LetG⁰ be the graph obtained from Gby moving all the edges at vi except uvi fromvi toufor eachi= 1,2. Thenρ(G)> ρ(G⁰).

Proof. Fori= 1,2, letS_i ={z ∈V(G₂)\ {vi}:d_G2(u, z)−d_G2(v_i, z) = 1}. As deg_G2(v_i)≥2, one hasS_i6=∅ fori= 1,2. AsdG₂(v1, w)6=dG₂(v2, w) for anyw∈V(G2)\ {u}, one hasS1∩S2=∅.

Choosew∈NG₁(u). Letx=x(G⁰), then by Lemma3.1, we havexw+xu−xv_i>0 fori= 1,2.

As we pass fromGtoG⁰, fori= 1,2, the distance between a vertex ofSi and a vertex ofV(G1) is decreased by 1, the distance between a vertex of Si and vi is increased by 1, and the distance between any other vertex pair is decreased or remains unchanged. So

1

2(ρ(G)−ρ(G⁰))≥ 1

2x^>(D(G)−D(G⁰))x

≥

2

X

i=1

σ_G⁰(S_i) (σ_G⁰(G₁)−x_vi)

≥

2

X

i=1

σ_G⁰(S_i) (x_w+x_u−x_vi)

>0,

and thusρ(G)> ρ(G⁰).

The following is Lemma 3.4 of [3].

Corollary 3.8. Let G be a connected graph with a chain C of odd length `, where ` ≥5. Let uv₁ and uv₂ be two edges on the chain C. Suppose that deg_G(u) ≥ 3. If G⁰ = G− {v1w : w ∈ NG(v1)\ {u}} − {v2w : w ∈ NG(v2)\ {u}}+{uw:w∈(NG(v1)∪NG(v2))\ {u}}, thenρ(G)> ρ(G⁰).

Proof. LetG1 be the component ofG−E(C) containingu, andG2 be the subgraph ofGinduced byV(G)\ (V(G₁)\ {u}). Letw∈V(G₂)\ {u}. Denote byzthe vertex onCsuch that its distance towis minimum among all vertices ofC. It is evident thatz=wifwlies onC. Thend_G(v₁, w)−d_G(v₂, w) =d_G(v₁, z)−d_G(v₂, z). As C is a chain, the shortest path connecting v1 (v2, respectively) andz contains only vertices on C. Let P (Q, respectively) be the shortest path connectingv1(v2, respectively) andzinG. IfP andQare edge disjoint, then dG(v1, z) +dG(v2, z) =`−2, and as`is odd, we havedG(v1, z)6=dG(v2, z). Otherwise,|dG(v1, z)−dG(v2, z)|= d_G(v₁, v₂) = 2, so d_G(v₁, z) 6= d_G(v₂, z). In either case, d_G(v₁, w) 6= d_G(v₂, w). So the result follows from

Theorem3.7.

Corollary 3.9. Let H be a graph consisting of two nontrivial connected graphs H1 and H2 sharing a unique vertex usuch that E(H) =E(H1)∪E(H2). Suppose that uv1, . . . , uvk are pendant edges in H1, where k ≥2 and that NH₂(u) =N1∪N2, whereN1, N26=∅ andN1∩N2=∅. Let

G=H− {uw:w∈NH₂(u)}+{v_k−1w:w∈N1}+{vkw:w∈N2}.

For any vertexw∈V(H2)\ {u}, if all the paths fromutowwith lengthdG(u, w)pass only throughN1 or pass only throughN2, thenρ(G)> ρ(H).

Proof. Let G₁ =H₁− {v_k−1, v_k} and let G₂ be the subgraph of G induced by V(H₂)∪ {v_k−1, v_k}. Suppose thatz∈V(G2)\ {u},i.e.,z∈(V(H2)\ {u})∪ {vk−1, vk}. Note that any shortest path fromutoz inH2 goes through only vertices in N1 or N2. Correspondingly, any shortest path fromu to z in G2 goes through only vertices inN1or N2. Consequently,dG(v_k−1, z)6=dG(vk, z). Now by Theorem3.7,ρ(G)> ρ(H).

We remark that Corollary3.9and Theorem3.7are equivalent. Ifk≥3, then Corollary3.9is just Theorem 2.3 of [16].

4. Graphs minimizing the distance spectral radius

First we determine the graphs that minimize the distance spectral radius among all homeomorphically irre- ducible unicylic graphs onn≥6 vertices.

Lemma 4.1([9]). Fork≥2 and 1≤a1 ≤a2−2, letG be a graph obtained from a connected graph G0 with two vertices u1 andu2 such thatNG₀(u1)\ {u2} ⊆NG₀(u2)\ {u1}, by attaching ai pendant vertices to ui for each i = 1,2. Let G⁰ be the graph obtained from G by moving one pendant edge at u2 from u2 to u1. Then ρ(G)< ρ(G⁰).

LetUn be a unicylic graph onnvertices obtained from a triangleK3withV(K3) ={v1, v2, v3}, by attaching a pendant vertex tovi fori= 1,2, respectively, and attachingn−5 pendant vertices tov3.

Theorem 4.2. Let Gbe a homeomorphically irreducible unicylic graph onn≥6 vertices. Then ρ(G)≥ρ(Un) with equality if and only if G∼=Un.

Proof. LetGbe a homeomorphically irreducible unicylic graph onnvertices that minimizes the distance spectral radius.

Let g be the girth of the unique cycle C of G. Let u be a vertex on C. Since G is a homeomorphically irreducible unicylic graph, we have deg_G(u)≥3. Letv1, v2 be two neighbors ofuonC.

Suppose that g≥4. Suppose that g is even. LetG⁰ be the graph obtained from Gby moving all the edges atv₁ exceptuv₁ fromv₁ tou. Note thatG⁰ is a homeomorphically irreducible unicylic graph onnvertices. By Theorem3.2or Corollary3.4,ρ(G⁰)< ρ(G), a contradiction. Thusg is odd. LetG⁰⁰be the graph obtained from Gby moving all the edges atvi exceptuvi from vi toufor eachi= 1,2. Obviously,G⁰⁰is a homeomorphically irreducible unicylic graph on n vertices. By Theorem 3.7 or Corollary 3.8, we have ρ(G⁰⁰) < ρ(G), also a contradiction. It thus follows thatg= 3.

Suppose thatGhas an edge, sayvw, outsideCthat is not a pendant edge. Evidently,vwis a cut edge ofG.

LetG^∗be the graph obtained fromGby moving all the edges atwexceptvwfromwtov. It is obvious thatG^∗ is a homeomorphically irreducible unicylic graph onnvertices. By Theorem3.2or Corollary3.3,ρ(G^∗)< ρ(G), a contradiction. Thus, every edge ofG outsideC is a pendant edge. That is,Gis a unicylic graph obtainable from a triangle K₃ with V(K₃) = {v1, v₂, v₃} by attaching a_i pendant vertices to v_i for i = 1,2,3, where 1≤a1≤a2≤a3.

Ifn= 6,7, thenG∼=Un.

Suppose that n≥ 8 and a₂ ≥ 2. Let Ge be the graph obtained from G by moving one pendant edge at v₂ from v2 to v3. Obviously, Ge is a homeomorphically irreducible unicylic graph on n vertices. By Lemma 4.1, ρ(G)e < ρ(G), a contradiction. So a2= 1. That is,a1=a2= 1 anda3=n−5,i.e.,G∼=Un. In the following, we determine the trees that minimize the distance spectral radius among all trees on n vertices with given number of vertices of degree two.

Let G be a connected graph with v ∈ V(G). Fork, ` ≥ 0, let G(v, k, `) be the graph obtained from G by attaching two pathsPk andP`at one end vertices tov. The following lemma was established in [13], for which a simple argument was given in [15].

Lemma 4.3. LetGbe a connected graph withv∈V(G). Ifk≥`≥1, thenρ(G(v, k, `))< ρ(G(v, k+ 1, `−1)).

A tree is called starlike if it has exactly one vertex of degree at least three; this vertex is called the branching vertex. If the branch vertex has degree s, we call it an s-starlike tree. For an s-starlike tree T on n vertices with branching vertexu, each path connectinguand a pendant vertex is called a leg. Denote bya1, . . . , asthe lengths of theslegs ofT. Thena1+· · ·+as=|E(T)|=n−1. Assume thata1≥ · · · ≥as. Ifa1−as= 0,1, then the multiset {a₁, . . . , a_s} composes of bⁿ⁻¹_s c+ 1 with multiplicity r andbⁿ⁻¹_s cwith multiplicity s−r, where r=n−1−sbⁿ⁻¹_s c. In this case, we call it ans-starlike tree of almost equal leg lengths, denoted by S_n,s.

Let T be a tree on n vertices with t vertices of degree two. If t = n−2, then T ∼= P_n. Note that t = n−3 is impossible, since T has at least two pendant vertices, and the remaining unique vertex has degree 2(n−1)−2(n−3)−1·2 = 2.

Theorem 4.4. Let T be a tree onn vertices with t vertices of degree two, where0≤t≤n−4. Thenρ(T)≥ ρ(Sn,n−t−1)with equality if and only if T ∼=Sn,n−t−1.

Proof. LetT be a tree on nvertices witht vertices of degree two that minimizes the distance spectral radius.

Since 0≤t≤n−4, the maximum degree ofT is at least three.

Suppose that there are at least two vertices of degree at least three inT. Then we choose two such vertices, sayuandv, by requiring that the distance between them is as small as possible. LetP be the path connecting uandv. Ifuandvare not adjacent, then each vertex onP exceptuandv has degree two. Letwbe the vertex adjacent to v on P (w = uif u and v are adjacent). Let T⁰ be the tree obtained from T by moving all the edges at v exceptwv from v tou. It is easily seen thatT⁰ possessest vertices of degree two. By Theorem3.6, ρ(T⁰)< ρ(T), a contradiction. Thus,T has exactly one vertex of degree at least three. That is,T is ans-starlike tree for somes. Assumea1, . . . , as are the lengths of the legs, wherea1≥ · · · ≥as. ThenPs

i=1ai=n−1 and Ps

i=1(a_i−1) =t. Sos=n−t−1. By Lemma4.3,a₁−a_n−t−1= 0,1. That is,T is an (n−t−1)-starlike tree

of almost equal leg lengths, orT ∼=Sn,n−t−1.

Acknowledgements. We thank the referees for careful reading and constructive comments on the manuscript. This work was supported by the Youth Innovative Talent Project of Guangdong Province of China (No. 2020KQNCX160) and the Basic and Applied Basic Research Project of Guangzhou Municipality of China (No. 202102080685).

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