HEAT KERNELS ON FORMS DEFINED ON A SUBGRAPH OF A COMPLETE GRAPH

COMPLETE GRAPH

YONG LIN, SZE-MAN NGAI, AND SHING-TUNG YAU

Abstract. We study the heat kernel expansion of the Laplacian onn-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.

1. Introduction

Given a directed complete graph K and a subgraph G, one can define n-forms on both G and K as well as Laplacians on these forms (see e.g., [6, 7]). The main purpose of this paper is to derive formulas for the the heat kernel expansion for the Laplacian on n-forms on G in terms of the heat kernel of the Laplacian on n-form on K. This has an analog in Riemannian geometry, with K playing the role of the Euclidean space Rⁿ and G playing the role of an n-dimensional Riemannian manifold. Spectral graph theory and heat heat kernels on graphs have been studied by many authors (see [3] and the references therein). In general, it is not easy to compute the heat kernel on graphs. Nevertheless, Chung and Yau [4, 5] derived formulas for the heat kernel on lattice graphs, n-cycles, k-regular graphs, and the k-tree. Grigor’yan and Telcs [8] obtained conditions under which a two-sided sub-Gaussian heat kernel estimate for a weighted graph holds.

More recently, Chinta et al. [2] derived a formula for the heat kernel on regular trees in terms of the classicalI-Bessel functions.

This paper studies heat kernels on subgraphs. In addition to functions on the graphs and subgraphs, we also study forms on them. In the classical setting, the heat kernels on forms yields interesting geometric information, such as the Euler characteristic and the Gauss-Bonnet Theorem (see [11, 14]). For non-directed graphs, the Gauss-Bonnet Theorem and Mckean-Singer Theorem on the Euler characteristic have been studied by Knill [9, 10]. In this paper, we consider directed graphs. We start by deriving a formal expression for the heat kernel of the Laplacian on n-forms on a subgraph in terms of a geometric series of an operator and the heat kernel onn-forms defined

Date: September 29, 2018.

2010Mathematics Subject Classification. Primary: 05C50, 35K08; Secondary:35K05, 39A12.

Key words and phrases. Laplacian; heat kernel;n-forms; complete graph.

The first two authors are supported in part by the National Natural Science Foundation of China, grants 11671401, 11771136, and 11271122. The first two authors were supported in part by the Center of Mathematical Sciences and Applications (CMSA) of Harvard University. The second author is also supported by Construct Program of the Key Discipline in Hunan Province and the Hunan Province Hundred Talents Program.

1

on the entire graph, namely,

H_t^G(x, y) = X^∞

m=0

T^m

H_t^K(x, y), (1.1)

whereT is some linear operator. See Theorem 2.4 in Section 2.

For 0-forms, i.e., functions defined onK, the termsT^mH_t^Kin (1.1) can be computed explicitly in terms of the difference of the Laplacians ∆^Gc := ∆^K−∆^G, where ∆^Kand ∆^Gare the combinatorial Laplacians on K and Grespectively. This allows us to derive a formula for H_t^G(x, y) and express it in three different forms. See Theorem 3.4, Corollary 3.4, and Corollary 3.7 in Section 3.

Section 4 is devoted to computing the coefficients of the subgraph heat kernel expansion obtained in Section 3.

In Section 5, we use another method to obtain the expansion of the heat kernel on a subgraph of a complete graph. This method was first introduced by Minakshisundaram-Pleijel and then used by Minakshisundaram to construct the heat kernel on compact Riemannian manifolds (see [1, 12, 13]), which has the form:

1

(4πt)^n/2e^{−d(x,y)2/(4t)} u0(x, y) +u1(x, y)t+· · ·+u_k(x, y)t^k+· · · .

We compute explicit formulas for the functions that are analogs of u_i(x, y),i= 0,1,2 (see Propo- sition 5.1). The terms in the series expansion of a subgraph heat kernel obtained by the two different methods appear in quite different forms. We verify that the first few terms of the two series expansions agree.

Fornforms it is in general not clear how T^mH_t^K(x, y) can be computed explicitly. Nevertheless, we show Section 6 that by solving a system of ODEs, one can derive an explicit formula for the heat kernel on a 1-forms on a complete graph, and use it to obtain an expression forH_t^G(x, y).

2. Recursive formula for heat kernels on n-forms of a subgraph

Let K =K_N be the complete graph with N vertices. Let V₀ :={1,2, . . . , N} denote the set of vertices. For n≥1, let

Vn:={i₀· · ·in:i0, . . . , in∈V0, ij 6=ij+1 for all j= 0, . . . , n−1}

denote the set of directed paths of lengthn+ 1. LetGbe a subgraph ofK with vertex setV₀^G⊆V0

and edge set V₁^G⊆V1. Forn≥1, let

V_n^G:={i₁· · ·in∈Vn:ijij+1 ∈V₀^G for allj = 1, . . . , n−1}

denote the set of directed paths in the graphG.

We let G^cbe the complement of Gdefined as follows. LetV₀^Gc :=V₀\V₀^G and call it the set of vertices of G^c. LetV₁^Gc :=V1\V₁^G. Note that G^cis not necessarily a graph, since an edge in V₁^Gc

does not necessarily connect two vertices inV₀^Gc. In fact, in Example 2.1 below,G^chas vertex set V₀^Gc ={3}and edge set V₁^Gc ={13,23,31,32} and thus it is not a graph. For eachn≥1, let

V_n^Gc :=Vn\V_n^G (2.1)

be the set of directed paths of length n+ 1 associated withG^c. Note that a directed path in V_n^Gc may contain a subpath that belongs to some V_k^G, 1 ≤ k ≤ n−1. For instance, in Example 2.1 below, the path 123∈V₂^Gc contains the subpath 12 that belongs toV₁^G.

For eachn≥0, we call any real-valued function onVnann-form onVn, and let Λⁿbe the vector space of all n-forms on V_n. Let {e^i0···in}_{i0···in∈Vn} be the canonical basis on Λⁿ with e^i0···in taking the value 1 at i0· · ·in and zero elsewhere. Define the exterior operator dn = d^K_n : Λⁿ → Λⁿ⁺¹ as follows. For

ω= X

i0···in∈Vn

ωi0···ine^i0···in ∈Vn, (2.2)

define

(d_nω)_i0···in+1 :=

n+1

X

k=0

(−1)^kω_i

0···ˆik···in+1,

where ˆi_k means that the indexi_kis removed. For eachn≥0, we also defined^G_n and d^G_n^c as follows.

Letω be as in (2.2). Then

d^G_n(ω) := X

i0···in∈Vn

ω_i0···ind^G_n(e^i0···in),

where

d^G_n(e^i0···in) :=

(d_n(e^i0···in) ifi₀· · ·i_n∈V_n^G,

0 otherwise. (2.3)

Similarly, define

d^G_n^c(ω) := X

i0···in∈Vn

ωi0···ind^G_n^c(e^i0···in),

where

d^G_n^c(e^i0···in) :=

(dn(e^i0···in) ifi0· · ·in∈V_n^Gc,

0 otherwise. (2.4)

It follows directly from the above definitions that

d_n=d^G_n +d^G_n^c. (2.5)

Example 2.1. Consider the complete graph K₃ with vertices {1,2,3}. Let G be the complete subgraph with vertices{1,2}. Then

V₀^G={1,2}, V₁^G={12,21}, V₂^G={121,212}, V₀^Gc ={3}, V₁^Gc ={13,23,31,32},

V₂^Gc =V₂^K\V₂^G={123,131,132,213,231,232,312,313,321,323}.

d^G₀ =

















−1 1 0 1 −1 0

0 0 0

0 0 0

0 0 0

0 0 0

















and d^G₀^c =

















0 0 0

0 0 0

−1 0 1

0 −1 1

1 0 −1

0 1 −1















 .

Notice that d0 =d^G₀ +d^G₀^c.

Let ∆0 := (d^K₀ )^∗d^K₀ be the Laplacian on 0-forms, where A^∗ denotes the transpose of A. For n≥1, let

∆^K_n := (d^K_n)^∗d^K_n +d^K_n−1(d^K_n−1)^∗ (2.6) be the Laplacian onn-forms. Define ∆^G_n and ∆^G_n^c analogously.

Proposition 2.2. The following relations hold.

(a) For any n≥0,

(d^G_n)^∗(d^G_n^c) = 0 and (d^G_n^c)^∗(d^G_n) = 0. (2.7) (b) ∆^K₀ = ∆^G₀ + ∆^G₀^c.

(c) For any n≥1,

∆^K_n = ∆^G_n + ∆^G_n^c+d^G_n−1(d^G_n−1^c )^∗+d^G_n−1^c (d^G_n−1)^∗. (2.8) Proof. (a) Lete^i0···in be a canonical basis element of the vector space of alln-forms. Then by (2.3) and (2.4),

(d_ne^i0···in)_j0...jn+1 =

((d^G_ne^i0···in)j0...jn+1, ifj0. . . jn+1 ∈V_n^G,

(d^G_n^ce^i0···in)_j0...jn+1, ifj₀. . . j_n+1 ∈V_n^Gc. (2.9)

Now by (2.1), the non-zero rows of d^G_n^c are exactly the zero rows of d^G_n, i.e., the zero columns of (d^G_n)^∗. Hence the equalities in 2.7 follow.

(b) By (2.5) and (2.7), we have

(d^K_n)^∗d^K_n = (d^G_n +d^G_n^c)^∗(d^G_n +d^G_n^c) = (d^G_n)^∗d^G_n + (d^G_n^c)^∗d^G_n^c. (2.10) Also, by using (2.5), we have

(d^K_n−1)(d^K_n−1)^∗ = (d^G_n−1+d^G_n−1^c )(d^G_n−1+d^G_n−1^c )^∗

=d^G_n−1(d^G_n−1)^∗+d^G_n−1(d^G_n−1^c )^∗+ (d^G_n−1^c )(d^G_n−1)^∗+ (d^G_n−1^c )(d^G_n−1^c )^∗. (2.11) Thus, (2.8) follows by combining (2.10), (2.11), and the definitions of ∆^K_n,∆^G_n,∆^G_n^c.

To simplify notation we let

L−1:= 0 and Ln−1:=d^G_n−1(d^G_n−1^c )^∗+d^G_n−1^c (d^G_n−1)^∗ forn≥1. (2.12) Proposition 2.3. Let n≥0. Then for allx, y∈V_n and t≥s≥0,

H_t^G(x, y) =H_t^K(x, y)− Z t

0

H_t−s^K (∆^G_n^c+Ln−1)H_s^G

(x, y)ds.

Proof. First, by the fundamental theorem of calculus, Z t

0

∂

∂s X

z∈Vn

H_s^G(z, y)H_t−s^K (x, z)

ds=H_t^G(x, y)−H_t^K(x, y). (2.13)

Computing the derivative on the left-side of (2.13), and using the symmetry of the operators

∆^G_n,∆^G_n^c, andLn−1, we get

∂

∂s X

z∈Vn

H_s^G(z, y)H_t−s^K (x, z)

= X

z∈Vn

−H_t−s^K (x, z)∆^G_n,zH_s^G(z, y) +H_s^G(z, y)∆^K_n,zH_t−s^K (x, z)

= X

z∈Vn

−H_s^G(z, y)∆^G_n,zH_t−s^K (x, z) +H_s^G(z, y)∆^K_n,zH_t−s^K (x, z)

= X

z∈Vn

H_s^G(z, y)(∆^K_n,z−∆^G_n,z)H_t−s^K (x, z)

= X

z∈Vn

H_s^G(z, y)(∆^G_n,z^c +Ln−1,z)H_t−s^K (x, z) (by (2.8) and (2.12))

= X

z∈Vn

H_t−s^K (x, z)(∆^G_n,z^c +Ln−1,z)H_s^G(z, y)

=

H_t−s^K (∆^G_n^c +Ln−1)H_s^G

(x, y).

(2.14)

Combining (2.13) and (2.14) yields the desired equality.

LetF be the vector space of all real-valued functions on [0,∞)×V_n². LetT :F → F be a linear operator defined as

T f_t(x, y) :=− Z t

0

H_t−s^K (∆^G_n^c+Ln−1)

f_s(x, y)ds. (2.15)

Theorem 2.4. Let T be defined as in (2.15) and assume thatkTk<1. Then H_t^G(x, y) =

^∞ X

m=0

T^m

H_t^K(x, y).

Proof. By Proposition 2.3,

H_t^G(x, y) =H_t^K(x, y) +T H_t^G(x, y)

=H_t^K(x, y) +T

H_t^K(x, y) +T H_t^G(x, y)

=· · ·

= (I +T+T²+· · ·)H_t^K(x, y),

completing the proof.

3. Heat kernel on 0-forms

A complete graphKN hasN vertices and N(N−1)/2 edges. The combinatorial Laplacian has eigenvalues 0 (with multiplicity 1) and N (with multiplicity N −1). Let V be the set of vertices of K_N. Let G be a sub-graph of K_N. Let G^c denote the complement of G obtained by removing those edges in KN that appear in G.

Recall that the combinatorial Laplacian ∆ on a graph is defined as ∆ =A−D, whereAand D are the adjacency and degree matrices respectively. Let H_t^K(x, y), H_t^G(x, y), H_t^Gc(x, y) denote the combinatorial Laplacians corresponding to K, G, G^c respectively. We use similar notation for the Laplacian ∆ and the degree dx of an element. Then

∆^K = ∆^G+ ∆^Gc.

It is well known that that heat kernel associated to ∆^K is given by H_t^K(x, y) =

(1

N + (1−_N¹)e^{−N t}, x=y,

1

N − _N¹e^{−N t}, x6=y. (3.1)

It can be obtained by expressing H_t^K(x, y) in terms of the eigenfunctions and eigenvalues of ∆^K.

Proposition 3.1. For all x, y∈V and t≥0, H_t^G(x, y) =H_t^K(x, y)−e^{−N t}

Z t 0

e^{N s}∆^G_x^cH_s^G(x, y)ds.

Proof. By using the proof of Proposition 2.3, we have H_t^G(x, y)−H_t^K(x, y) =

Z t 0

X

z∈V

H_s^G(z, y)∆^G_z^cH_t−s^K (x, z)ds. (3.2)

In view of (3.1), the integrand in (3.2) is equal to X

z∈V

X

w∼

Gcz

h

H_t−s^K (x, w)−H_t−s^K (x, z)i

H_s^G(z, y)

= X

w∼

Gcx

h

H_t−s^K (x, w)−H_t−s^K (x, x) i

H_s^G(x, y) +

X

w∼

Gcz,z6=x

h

H_t−s^K (x, w)−H_t−s^K (x, z) i

H_s^G(z, y)

= X

w∼

Gcx

−e^−N(t−s)H_s^G(x, y) + X

w∼

Gcz

h

H_t−s^K (x, x)−H_t−s^K (x, z) i

H_s^G(z, y)

= X

w∼

Gcx

−e^−N(t−s)H_s^G(x, y) +X

x∼

Gcz

e^−N(t−s)H_s^G(z, y)

=−e^−N(t−s)d^G_x^cH_s^G(x, y) +X

x∼

Gcz

e^−N(t−s)H_s^G(z, y)

=−e^−N(t−s)

d^G_x^cH_s^G(x, y)− X

x∼

Gcz

H_s^G(z, y)

=−e^−N(t−s)∆^G_x^cH_s^G(x, y).

Substituting this into (3.2), we obtain the desired formula.

Now let F be the space of all real-valued functions on [0,∞)×V. Let T :F → F be a linear operator defined as

T u(t, x) :=−e^{−N t} Z t

0

e^{N s}∆^G_x^cu(s, x)ds.

Let

kuk_∞= supn

|u(t, x)|:x∈V, t∈[0,∞)o . Proposition 3.2. If t <(1/N) log(2(N −1)/(N −2)), then kTk<1.

Proof. It follows from definitions that T u(t, x) =−e^{−N t}

Z t 0

e^{N s} X

w∼

Gcx

[u(s, w)−u(s, x)]ds.

Hence

kT u(t, x)k ≤e^{−N t} Z t

0

e^{N s}·2(N −1)kuk_∞ds= 2(N−1)

N (1−e^{−N t})kuk_∞.

Now, solving the inequality 2(N −1)(1−e^{−N t})/N < 1 yields the stated upper bound fort below

which one haskTk<1.

Under the hypothesis of Proposition 3.2,kTk<1 and thus by using Proposition 3.1,

H_t^G(x, y) = (I+T+T²+· · ·)H_t^K(x, y). (3.3) To derive a more explicit formula for H_t^G(x, y), for eachy ∈V, we let u_y :V →V be the function defined as

u^G_y^c(x) :=











d^G_x^c ifx=y,

0 ifx6=y and x∼

G y,

−1 ifx6=y and x6∼

G y.

(3.4)

Theorem 3.3. Let u^G_y^c :V →V be defined as in (3.4). Then for any x, y∈V, and all t≥0,

H_t^G(x, y) =H_t^K(x, y) +te^{−N t}u^G_y^c(x) +e^{−N t}

∞

X

m=2

(−1)^m−1t^m m!

∆^G_x^cm−1

u^G_y^c(x)

=











1

N −_N¹e^{−N t}+e^{−N t}P∞ m=1

(−1)^m−1t^m m!

∆^G_x^cm−1

u^G_y^c(x), y 6=x,

1

N + (1−_N¹)e^{−N t}+e^{−N t}P∞ m=1

(−1)^m−1t^m m!

∆^G_x^cm−1

u^G_y^c(x), y =x,

(3.5)

where

∆^G_x^cm−1

denotes the (m−1)-fold composition of∆^G_x^c (or equivalently, the(m−1)th power of ∆^G_x^c).

Proof. We first compute T H_t^G(x, y) by considering the following three cases. Throughout the calculations below, we denotex ∼

G^c y(i.e.,x andy are neighbors in the graphG^c) simply byx∼y.

Case 1. x=y. Then

T H_t^K(x, y) =T H_t^K(x, x) =−e^{−N t} Z t

0

e^{N s}∆^G_x^cH_s^K(x, x)ds

=−e^{−N t} Z t

0

e^{N s}X

w∼x

h

H_s^K(w, x)−H_s^K(x, x)i ds

=−e^{−N t} Z t

0

e^{N s}X

w∼x

(−e^{−N s})ds

=−e^{−N t} Z t

0

e^{N s}(−e^{−N s})d^G_x^cds

=te^{−N t}d^G_x^c.

Case 2. x6=y and x∼y. Then T H_t^K(x, y) =−e^{−N t}

Z t 0

e^{N s}∆^G_x^cH_s^K(x, y)ds

=−e^{−N t} Z t

0

e^{N s}X

w∼x

h

H_s^K(w, y)−H_s^K(x, y) i

ds

=−e^{−N t} Z t

0

e^{N s}n X

w∼x,w6=y

h

H_s^K(w, y)−H_s^K(x, y)i

+ X

w∼x,w=y

h

H_s^K(w, y)−H_s^K(x, y)io ds

=−e^{−N t} Z t

0

e^{N s}(0 +e^{−N s})ds

=−te^{−N t}.

Case 3. x 6=y and x 6∼y. Then, unlike in Case 2, the situation w∼x and w =y cannot occur.

Hence

T H_t^K(x, y) =−e^{−N t} Z t

0

e^{N s}

X

w∼x,w6=y

h

H_s^K(w, y)−H_s^K(x, y) i

ds

=−e^{−N t} Z t

0

e^{N s}·0ds

= 0.

Thus,

T H_t^K(x, y) =te^{−N t}u^G_y^c(x). (3.6) Now we can expressT²H_t^K(x, y) conveniently as

T²H_t^K(x, y) =−e^{−N t} Z t

0

e^{N s}∆^G_x^c

T H_s^K(x, y)

ds

=−e^{−N t} Z _t

0

e^{N s}∆^G_x^c se^{−N s}u^G_y^c(x)

ds (by (3.6))

=−e^{−N t} Z t

0

s∆^G_x^cu^G_y^c(x)ds

=−t²e^{−N t}

2! ∆^G_x^cu^G_y^c(x).

By induction and a similar derivation, for all m≥2, T^mH_t^K(x, y) = (−1)^m−1t^me^{−N t}

m!

∆^G_x^cm−1

u^G_y^c(x). (3.7)

Combining this with equation (3.3) proves the proposition.

By using (3.1), we can rewrite the formula forH_t^G(x, y) in the following form.

Corollary 3.4. The formula in Proposition 3.3 can be expressed as

H_t^G(x, y) =















H_t^K(x, y)

1 +_eN t+N−1^N

P∞ m=1

(−1)^m−1t^m

m! (∆^G_x^c)^m−1u^G_y^c(x)

, y=x,

H_t^K(x, y)

1 +_eN t^N−1

P∞ m=1

(−1)^m−1t^m

m! (∆^G_x^c)^m−1u^G_y^c(x)

, y6=x.

We now study the radius of convergence of the power series in (3.5). We first prove a lemma.

Lemma 3.5. For any m≥1 and any vertices x, y∈V₀,

(∆^G_x^c)^m−1u^G_y^c(x)

≤2^m−1N^m.

Proof. The inequality clearly holds whenm= 1. Assume that it holds for somem≥1. Then

(∆xG^c)^mu^G_y^c(x) =

X

w∼

Gcx

h

(∆^G_w^c)^m−1u^G_y^c(w)−(∆x)^m−1u^G_y^c(x) i

≤ X

w∼

Gcx

2^m−1N^m+ 2^m−1N^m

(Induction hypothesis)

≤d^G_x^c2^mN^m

≤2^mN^m+1,

which completes the proof.

Note that the series in Corollary 3.4 can also be written as

H_t^G(x, y) =















H_t^K(x, y)

1 +_eN t^{N t}+N−1

P∞ m=1

(−1)^m−1t^m−1

m! (∆^G_x^c)^m−1u^G_y^c(x)

, y=x,

H_t^K(x, y)

1 +_eN t^{N t}−1

P∞ m=1

(−1)^m−1t^m−1

m! (∆^G_x^c)^m−1u^G_y^c(x)

, y6=x.

(3.8)

Proposition 3.6. Consider the expansion in Corollary 3.4.

(a) The radius of convergence of the series

∞

X

m=1

(−1)^m−1t^m m!

∆^G_x^c m−1

u^G_y^c(x) and

∞

X

m=1

(−1)^m−1t^m m!

∆^G_x^c m−1

u^G_y^c(x)

that appear in (3.5) and (3.8) is∞.

(b) The functions t/(e^{N t}+N −1) is real analytic in some open neighborhood of 0, and so is the function

f(t) :=





 t

e^{N t}−1 if t6= 0, 1

N if t= 0.

Proof. (a) By Stirling’s formula,m!∼cm^1/2+me^−m and hence ^m√

m!∼Cm for some constant C.

Thus, by Lemma 3.5, for all x, y∈V₀,

m→∞lim

m

r

(−1)^m−1t^m

m! (∆^G_x^c)^m−1u^G_y^c(x)

≤ lim

m→∞

m

r

|t|^m

m! 2^mN^m+1

= lim

m→∞

2N ^m√ N|t|

m√ m!

=0.

Hence the series converges for all t∈R. The proof of the second series is the same.

(b) The function t/(e^{N t}+N −1) is clearly real analytic in some open neighborhood on 0. We now consider f. First, it is clear the f is continuous at 0. Let f(z) be the extension of f to C. Then

f(z) = 1

NP∞ k=0

(N z)^k−1 k!

,

which is (complex) analytic on some open neighborhood of 0 in C. Thus, f(t) is real analytic on

some open neighborhood of 0 in R.

Using Proposition 3.6 and Taylor series expansion, we can rewrite the heat kernel H_t^G(x, y) in Corollary 3.4 in the following form.

Corollary 3.7. Let u^G_y^c(x) defined in (3.4). Then (a) If y=x, then

H_t^G(x, y) =H_t^K(x, y) 1 +d^G_x^ct−1 2

2d^G_x^c+ ∆^G_x^cu^G_y^c(x) t² +1

6

(6−3N)d^G_x^c+ 3∆^G_x^cu^G_y^c(x) + (∆^G_x^c)²u^G_y^c(x)

t³ + 1

12

(−12 + 12N −2N²)d^G_x^c−(6−3N)∆^G_x^cu^G_y^c(x)−(∆^G_x^c)²u^G_y^c(x)

t⁴ +O(t⁵)

! .

(b) If y6=x and y∼

G x, then H_t^G(x, y) =H_t^K(x, y) 1−1

2∆^G_x^cu^G_y^c(x)t+ 1 12

3N∆^G_x^cu^G_y^c(x) + 2(∆^G_x^c)²u^G_y^c(x)

t²

− 1 24

N²∆^G_x^cu^G_y^c(x) + 2N(∆^G_x^c)²u^G_y^c(x) t³ + 1

72N²(∆^G_x^c)²u^G_y^c(x)t⁴+O(t⁵)

! .

(c) If y6=x and y6∼

G x, then H_t^G(x, y) =H_t^K(x, y) 1

2

N−∆^G_x^cu^G_y^c(x)

t+ 1 12

−N²+ 3N∆^G_x^cu^G_y^c(x) + 2(∆^G_x^c)²u^G_y^c(x)

t²

− 1 24

N²∆^G_x^cu^G_y^c(x) + 2N(∆^G_x^c)²u^G_y^c(x) t³ + 1

720

N⁴+ 10N²(∆^G_x^c)²u^G_y^c(x)

t⁴+O t⁵

! .

4. Computing the coefficients in the heat kernel expansion

This section is devoted to the computation of the coefficients in the heat kernel expansion of H_t^G(x, y). Let ck(x, y) be the coefficients oft^k in the above expression forH_t^G(x, y). Then we see from the expression for H_t^G(x, y) in Corollary 3.7 that

c0(x, y) :=











1 ifx=y,

1 ifx6=y and x∼

G y, 0 ifx6=y and x6∼

G y.

To compute the other coefficients, we letη^G,G(x, y) be the number ofG-neighbors of x that are alsoG-neighbors ofy, and letη^G,Gc(x, y) be the number ofG-neighbors ofxthat areG^c-neighbors ofy. Similarly we defineη^Gc,G(x, y) andη^Gc,Gc(x, y). We first establish some elementary properties relating these quantities andd^G_x,d^G_x^c.

Since K is a complete graph, we have

d^G_x +d^G_x^c =N −1. (4.1)

The following elementary identities follow immediately from symmetry:

η^G,Gc(y, x) =η^Gc,G(x, y), η^G,G(y, x) =η^G,G(x, y)

η^Gc,G(y, x) =η^G,Gc(x, y), η^Gc,Gc(y, x) =η^Gc,Gc(x, y). (4.2)

Proposition 4.1. (a) If x6=y and x∼

G y, then

d^G_x −1 =η^G,G(x, y) +η^G,Gc(x, y), (4.3) d^G_x^c =η^Gc,G(x, y) +η^Gc,Gc(x, y). (4.4) (b) If x6=y and x6∼

G y, then

d^G_x =η^G,G(x, y) +η^G,Gc(x, y), (4.5) d^G_x^c−1 =η^Gc,G(x, y) +η^Gc,Gc(x, y). (4.6) Proof. We only prove (b); the proof of (a) is similar. In this case, y is a G^c-neighbor of x. G- neighbors ofx can be partitioned into two classes: those that areG-connected toy and those that areG^c-connected to y. This gives rise to (4.5).

As for (4.6), we note thaty is G^c-neighbors of x. The remaining d^G_c^c−1G^c-neighbors of xcan be partitioned into two classes: those that are G-connected toy and those that are G^c-connected

toy. This proves (4.6).

Proposition 4.2. (a) If x6=y and x∼

G y, then

η^Gc,Gc(x, y) =η^G,G(x, y)−d^G_x +d^G_y^c+ 1

=η^G,G(x, y) +d^G_x^c−d^G_y + 1

=N+η^G,G(x, y)−d^G_x −d^G_y.

(4.7)

(b) If x6=y and x6∼

G y, then

η^Gc,Gc(x, y) =η^G,G(x, y)−d^G_x +d^G_y^c−1

=η^G,G(x, y) +d^G_x^c−d^G_y −1

=N−2−d^G_x −d^G_y +η^G,G(x, y).

(4.8)

Proof. We will prove (a); the proof of (b) is similar. Interchanging the roles ofx andy in (4.3) and using (4.2), we have

d^G_y −1 =η^G,G(y, x) +η^G,Gc(y, x) =η^G,G(x, y) +η^Gc,G(x, y),

d^G_y^c =η^Gc,Gc(y, x) +η^Gc,G(y, x) =η^Gc,Gc(x, y) +η^G,Gc(x, y). (4.9) Substituting these into (4.3) and (4.4) respectively, we get

d^G_x −1 =η^G,G(x, y) +d^G_y^c−η^Gc,Gc(x, y),

d^G_x^c =η^Gc,Gc(x, y) +d^G_y −η^G,G(x, y)−1, (4.10)

which imply the first two inequalities in (4.7), respectively. The third equality in (4.7) follows by

using (4.2).

Proposition 4.3.

∆^G_x^c

u^G_y^c(x)

=











−d^G_x^c −(d^G_x^c)², if x=y,

−η^Gc,Gc(x, y), if x6=y andx∼

G y,

−η^Gc,Gc(x, y) +d^G_x^c+d^G_y^c, if x6=y andx6∼

G y,

=











−(N −1−d^G_x)−(N −1−d^G_x)², if x=y,

−N −η^G,G(x, y) +d^G_x +d^G_y, if x6=y andx∼

G y, N −η^G,G(x, y), if x6=y andx6∼

G y.

Proof. Note that

∆^G_x^c

u^G_y^c(x)

= (∆^K_x −∆^G_x)

u^G_y^c(x)

= X

w∼

Kx

u^G_y^c(w)−u^G_y^c(x)

−X

w∼

Gx

u^G_y^c(w)−u^G_y^c(x)

= X

w∼

Gcx

u^G_y^c(w)− X

w∼

Gcx

u^G_y^c(x).

(4.11)

We divide the proof into three cases.

Case 1. y=x. In view of the definition ofu^G_y^c(x) and (4.11), we have

∆^G_x^c

u^G_y^c(x)

= X

w∼

Gcx

(−1)− X

w∼

Gcx

d^G_x^c

=−d^G_x^c−(d^G_x^c)²

=−(N−1−d^G_x)−(N−1−d^G_x)² (by (4.1)).

Case 2. y6=x andy ∼

G x. Using (4.11) followed by Proposition 4.2(a), we have

∆^G_x^c

u^G_y^c(x)

=

X

w∼

Gcx,w∼

Gcy

u^G_y^c(w) + X

w∼

Gcx,w6 ∼

Gcy

u^G_y^c(w)

−d^G_x^cu^G_y^c(x)

= X

w∼

Gcx,w∼

Gcy

(−1) + 0−0

=−η^Gc,Gc(x, y)

=−N −η^G,G(x, y) +d^G_x +d^G_y (by Proposition 4.2(a)).

Case 3. y6=x andy 6∼

G x. Using (4.11) followed by Proposition 4.2(b), we have

∆^G_x^c

u^G_y^c(x)

=

X

w∼

Gcx,w∼

Gcy

u^G_y^c(w) + X

w∼

Gcx,w6 ∼

Gcy

u^G_y^c(w)

+u^G_y^c(y)−d^G_x^cu^G_y^c(x)

= X

w∼

Gcx,w∼

Gcy

(−1) +d^G_y^c+d^G_x^c

=η^Gc,Gc(x, y) +d^G_y^c+d^G_x^c

=N −η^G,G(x, y) (by Proposition 4.2(b)).

Proposition 4.4.

(∆^G_x^c)²

u^G_y^c(x)

=



































2(d^G_x^c)²+ (d^G_x^c)³−P

w∼

Gcxη^Gc,Gc(w, x) +P

w∼

Gcxd^G_w^c, x=y, (d^G_x^c+d^G_y^c)η^Gc,Gc(x, y)−P

w∼

Gcx,w6=yη^Gc,Gc(w, y) +P

w∼

Gc

x,w6=y,w∼

Gcyd^G_w^c, x6=y and x∼

G y,

−(d^G_x^c)²−d^G_y^c−(d^G_y^c)²−d^G_x^cd^G_y^c+ (d^G_x^c+d^G_y^c)η^Gc,Gc(x, y)

−P

w∼

Gcx,w6=yη^Gc,Gc(w, y) +P

w∼

Gcx,w6=y,w∼

Gcyd^G_w^c, x6=y and x6∼

G y,

=







































−N²+N³−(1−3N+ 3N²)d^G_x −(2−3N)(d^G_x)²−(d^G_x)³

−P

w∼

Gcxη^G,G(w, x), x=y,

−2 + 2N + 2N²−(1 + 3N)d^G_x + (d^G_x)²−3N d^G_y + (d^G_y)²+d^G_xd^G_y

−(1−3N+d^G_x +d^G_y)η^G,G(x, y)

−P

w∼

Gcx,w6=y,w∼

Gcyη^G,G(w, y) +P

w∼

Gcx,w6=y,w∼_Gyd^G_w, x6=y and x∼

G y,

−N²+d^G_y −(3−3N+d^G_x +d^G_y)η^G,G(x, y)

−P

w∼

Gcx,w6=y,w∼

Gcyη^G,G(w, y) +P

w∼

Gcx,w6=y,w∼_Gy(d^G_w −η^G,G(w, y)), x6=y and x6∼

G y.

Proof.

(∆^G_x^c)²

u^G_y^c(x)

= X

w∼

Gcx

h

∆^G_w^c

u^G_y^c(w)

−∆^G_x^c

u^G_y^c(x) i

= X

w∼

Gcx

∆^G_w^c

u^G_y^c(w)

−d^G_x^c∆^G_x^c

u^G_y^c(x) .

(4.12)