Characteristic Power Series of Graph Limits

TL;DR

This paper introduces a characteristic power series for graphons, linking spectral properties to graph quasi-randomness and offering a new perspective on spectral theory for dense graph limits.

Contribution

It develops a characteristic power series for graphons as a limit of normalized reciprocal characteristic polynomials, providing a novel spectral characterization of graph quasi-randomness.

Findings

Defines a characteristic power series for graphons.

Connects the power series to the spectrum of the graphon.

Provides a new spectral perspective on graph limits.

Abstract

In this note, we show how to obtain a "characteristic power series" of graphons -- infinite limits of dense graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a new characterization of graph quasi-randomness and another perspective on spectral theory for graphons, a complete description of the function in terms of the spectrum of the graphon as a self-adjoint kernel operator. Interestingly, while we apply a standard regularization to classical determinants, it is unclear how necessary this is.

Equations

$$\sum _{v,w\in V(G)}\left||N_G(v)\cap N_G(w)|-n{p}^2\right|=o(n^3)$$

$$P_2(4)\iff P_2(t)\iff P_1(s)\iff P_1^{\prime }(s)\iff P_3\iff P_4\iff P_5\iff P_6$$

$$\psi_{\mathcal{G}}(z)=\left\{\begin{array}[]{ll}\lim_{n\rightarrow\infty}\left(x^{-n}\phi_{n}(x)\right)\Big{|}_{x=n/z}&\textrm{ if }z\neq 0\\ 1&\textrm{ otherwise },\end{array}\right.$$

$$\sum _{H\in {\mathcal{H}}_k}(-1{)}^{c(H)}{2}^{z(H)}[\#H\subseteq G]$$

$$(\#H\subseteq G)=[\#H\subseteq G]\cdot 2^{z(H)}\prod _i{a}_i^{m_i}{m}_i!$$

$$\eta (\lambda ):=\prod _i{b}_i^{m_i}{m}_i!$$

$$\frac{n!}{b_1{!}^{m_1}\cdots b_t{!}^{m_t}{m}_1!\cdots m_t!}=\frac{n!}{\eta (\lambda ){\prod }_{i=1}^t(b_i-1){!}^{m_i}}$$

$$x^{-n}{\varphi }_{G_n}(x)$$

$$=\sum _{k=0}^n\sum _{\lambda \in {\Lambda }_{n,k}^{\prime }}{x}^{-k}(-1{)}^k\frac{(\#{⨆}_i{C}_{{\lambda }_i}\subseteq G_n)}{\eta (\lambda )}$$

$$=\sum _{k=0}^n\sum _{\lambda \in {\Lambda }_{n,k}^{\prime }}{z}^k(-1{)}^k\frac{(\#{⨆}_i{C}_{{\lambda }_i}\subseteq G_n)}{n^k{\prod }_i{b}_i^{m_i}{m}_i!},$$

$$|{\psi }_n(z)|$$

$$=\sum _{k=0}^n\sum _{\lambda \in {\Lambda }_{n,k}^{\prime }}\prod _{i=1}^k\frac{|z{|}^{m_i}t(C_{b_i},G_n{)}^{m_i}}{b_i^{m_i}{m}_i!}$$

$$=\prod _{b\ge 2}\sum _{m\ge 0}\frac{|z{|}^{m}t(C_b,G_n{)}^m}{b^{m}m!}$$

$$=\prod _{b\ge 2}\exp \left(\frac{|z|t(C_b,G_n)}{b}\right)=\exp \left(\sum _{b\ge 2}\frac{|z|t(C_b,G_n)}{b}\right).$$

$$|{\psi }_n(z)|$$

$$={\left[\prod _i\left(1-{\mu }_i\right)\right]}^{-|z|}\cdot \exp \left(-|z|\sum _i{\mu }_i\right)={\left[\prod _i\left(1-{\mu }_i\right)\right]}^{-|z|}$$

$$\stackrel{(p)}{\det }(I-z\mathcal{A})=\prod _j\left[(1-{\lambda }_{j}z)\exp \left(\sum _{k=1}^{p-1}{\lambda }_j^k{z}^k/k\right)\right]$$

$${\psi }_{\mathcal{G}}(z)=\stackrel{(2)}{\det }(I-z\mathcal{G})\exp \left(z^2\cdot \frac{\Vert \mathcal{G}{\Vert }_2^2-\Vert \mathcal{G}{\Vert }_1}{2}\right).$$

$${\psi }_{\mathcal{G}}(z)=z^m\exp (a+bz+c{z}^2)\prod _r\left(1-\frac{z}{r}\right)\exp \left(\frac{z}{r}\right)$$

$${\psi }_n(z)$$

$$={\left(\det (I-A(G_n)x^{-1})\right)|}_{x=n/z}$$

$$=\det (I-A_n^{\prime }z).$$

$$\prod _j\left(1-{\lambda }_j(A_n^{\prime })z\right)\exp \left(\sum _{k=1}^3\frac{{\lambda }_j(A_n^{\prime }{)}^k{z}^k}{k}\right)\to \prod _j\left(1-z{\lambda }_j(\mathcal{G})\right)\exp \left(\sum _{k=1}^3\frac{{\lambda }_j(\mathcal{G}{)}^k{z}^k}{k}\right)$$

$$\stackrel{(2)}{\det }(I-z\mathcal{G})\prod _j\exp \left(\frac{{\lambda }_j(\mathcal{G}{)}^2{z}^2}{2}+\frac{{\lambda }_j(\mathcal{G}{)}^3{z}^3}{3}\right)$$

$${\psi }_n(z)\prod _j\exp \left(\frac{{\lambda }_j(A_n^{\prime }{)}^2{z}^2}{2}+\frac{{\lambda }_j(A_n^{\prime }{)}^3{z}^3}{3}\right)$$

$${\psi }_n(z)\exp \left(\frac{z^2|E(G_n)|}{n^2}\right)\to \stackrel{(2)}{\det }(I-z\mathcal{G})\exp \left(\frac{z^2\Vert \mathcal{G}{\Vert }_2^2}{2}\right)$$

$$\underset{n\to \infty }{\lim }{\psi }_n(z)=\stackrel{(2)}{\det }(I-z\mathcal{G})\exp \left(z^2\cdot \frac{\Vert \mathcal{G}{\Vert }_2^2-\Vert \mathcal{G}{\Vert }_1}{2}\right)={\psi }_{\mathcal{G}}(z)$$

$$W(x,y)=\left\{\begin{array}[]{ll}\mathcal{G}\left(\frac{x}{p},\frac{y}{p}\right)&\text{if }x,y\in[0,p)\\ \mathcal{H}\left(\frac{x-p}{1-p},\frac{y-p}{1-p}\right)&\text{if }x,y\in[p,1]\\ 0&\text{otherwise}\\ \end{array}\right..$$

$${\psi }_{\mathcal{G}{\oplus }_p\mathcal{H}}(z)={\psi }_{\mathcal{G}}(pz){\psi }_{\mathcal{H}}((1-p)z).$$

$$\stackrel{(2)}{\det }(I-z\mathcal{G}\oplus \mathcal{H})$$

$$=\stackrel{(2)}{\det }(I-zp\mathcal{G})\stackrel{(2)}{\det }(I-z(1-p)\mathcal{H})e^{-z{\sum }_i{\lambda }_i({\mathcal{G}}^{\prime }{\mathcal{H}}^{\prime })}.$$

$${\psi }_{\mathcal{G}\oplus \mathcal{H}}(z)$$

$$=\stackrel{(2)}{\det }(I-z{\mathcal{G}}^{\prime })\stackrel{(2)}{\det }(I-z{\mathcal{H}}^{\prime })$$

$$\cdot \exp \left[\frac{z^2}{2}\left(p^2\Vert \mathcal{G}{\Vert }_2^2+(1-p{)}^2\Vert \mathcal{H}{\Vert }_2^2-p^2\Vert \mathcal{G}{\Vert }_1-(1-p{)}^2\Vert \mathcal{H}{\Vert }_1\right)\right]$$

$$={\psi }_{{\mathcal{G}}^{\prime }}(z){\psi }_{{\mathcal{H}}^{\prime }}(z)={\psi }_{\mathcal{G}}(pz){\psi }_{\mathcal{H}}((1-p)z).$$

$${\psi }_{\mathcal{G}}(z)=\exp \left(\frac{(\Vert \mathcal{G}{\Vert }_2^2-\Vert \mathcal{G}{\Vert }_1^1)z^2}{2}+z\mathrm{tr}\mathcal{G}\right)\prod _{s\in \mathcal{S}}\left(1-sz\right),$$

Full text

††journal: European Journal of Combinatorics

1 Introduction

A research direction began in the 1980’s with graph quasi-randomness, extended through the 1990’s and early 2000’s with generalizations to non-uniform graph distributions and other combinatorial objects, became graph limit theory in the mid-2000’s, and culminated in Lovász’s now-canonical text [13]. The central idea is that, if a sequence of graphs $G_{n}$ with number of vertices tending to infinity has the property that the density of any particular subgraph tends to a limit, then $G_{n}$ itself tends to a limit object $\mathcal{G}$, called a “graphon”. There are several mutually (though non-obviously) equivalent ways to view graphons, and a central one is as a self-adjoint kernel operator from $L^{1}([0,1])$ to $L^{\infty}([0,1])$, an object type for which a well-established spectral theory exists. In particular, a graphon, when thought of as this kernel, is a symmetric function $[0,1]\times[0,1]\rightarrow[0,1]$, with two such kernels $U$ and $W$ “weakly isomorphic” (i.e., giving rise to the same graphon) if there are measure-preserving maps $f,g:[0,1]\rightarrow[0,1]$ so that $U(f,f)=W(g,g)$ almost everywhere. Indeed, we use the same notation throughout for $\mathcal{G}$ as well as (any representative of the equivalence class of) its kernel. There is also a key notion of subgraph density for graphons, with the property that densities in convergent sequences of graphs in a sequence converge to their densities in the limit graphon, often referred to as “left-convergence”. Furthermore, Szegedy ([17]) introduced a spectral theory of graphons by studying the eigenpairs of their kernels and showed that it is a natural analogue of the adjacency spectral theory of finite graphs. Here, we extend this perspective by showing that graphons are associated with a power series which is a certain normalized limit of the characteristic polynomial of graphs. Furthermore, we give an equivalent definition of this “characteristic power series” $\psi_{\mathcal{G}}(z)$ of a graphon $\mathcal{G}$ via a regularized determinant of its corresponding kernel function.

We also show that the characteristic power series can be used to characterize “quasi-randomness”. Suppose that $\{G_{n}\}_{n\geq 1}$ is a sequence of graphs with $|V(G_{n})|=n$. (In truth, all that is needed is that $|V(G_{n})|\rightarrow\infty$, but this is not more general.) We write $G=G_{n}$ for simplicity and $(\#H\subseteq G)$ for the number of labelled, not-necessarily induced copies of $H$ as subgraphs in $G$ (i.e., injective homomorphisms from $H$ to $G$). Then, by a classic 1989 paper of Chung, Graham, and Wilson ([5]), there is a large set of random-like properties (properties which hold asymptotically almost surely for graphs in the Erdős-Rényi model $G(n,p)$) which are mutually equivalent, and are therefore collectively referred to as (the sequence of graphs) $G$ being “quasi-random”111We state the version of this theorem for all densities $p\in[0,1]$, but it actually appeared first only for $p=1/2$.. Namely, let

$P_{1}(s)$ denote the property that the number of labelled occurrences of each graph on $s$ vertices as an induced subgraph of $G$ is $n^{s}(p^{|E(H)|}(1-p)^{\binom{s}{2}-|E(H)|}+o(1))$

• 2.

$P^{\prime}_{1}(s)$ denote the property that $(\#H\subset G)=n^{s}(p^{|E(H)|}+o(1))$ for each graph $H$ on $s$ vertices

• 3.

$P_{2}(t)$ denote the property that $|E(G)|\geq(p+o(1))n^{2}/2$ and $(\#C_{t}\subseteq G)\leq(1+o(1))(np)^{t}$, where $C_{t}$ is the $t$-cycle

• 4.

$P_{3}$ denote the property that $|E(G)|\geq(p+o(1))n^{2}/2$, $\lambda_{1}=pn(1+o(1))$, and $\lambda_{2}=o(n)$, where $|\lambda_{1}|\geq\cdots\geq|\lambda_{n}|$ are the complete set of adjacency eigenvalues of $G$ with multiplicity

• 5.

$P_{4}$ denote the property that, for all $S\subseteq V(G)$, $E(G[S])=p|S|^{2}/2+o(n^{2})$, where $G[S]$ denotes the subgraph of $G$ induced by $S$

• 6.

$P_{5}$ denote the property that, for all $S\subseteq V(G)$ with $|S|=\lfloor n/2\rfloor$, $E(G[S])=pn^{2}/8+o(n^{2})$, where $G[S]$ denotes the subgraph of $G$ induced by $S$

• 7.

$P_{6}$ denote the property that

[TABLE]

Theorem 1** (Chung-Graham-Wilson [5]).**

For $s\geq 4$ arbitrary and $t\geq 4$ even, and any fixed $p\in[0,1]$,

[TABLE]

If a graph sequence $G$ has these properties, it is called $p$-quasi-random, and these properties and any others also equivalent to them are known as ($p$-)quasi-random properties. Many other quasi-random properties have been added since to the list above, such as other families $\mathcal{F}$ of graphs whose occurrence as subgraphs at the “random-like rate” implies these properties (note that $\{K_{2},C_{4}\}$ is the “forcing” family given by $P_{2}(4)$), and also that $G_{n}$ converges to a $p$-constant graphon.

Here, we propose to add another property. First, given a left-convergent sequence of graphs $G$, let $\mathcal{G}$ be their limit, and let $\phi_{n}\in\mathbb{C}[x]$ denote the (adjacency) characteristic polynomials of $G_{n}$. Recall that the characteristic polynomial of a graph $G$ with adjacency matrix $A$ is defined to be $\phi(x)=\det(Ix-A)$; it is easy to see that $\phi$ is monic and has only real roots. Define

[TABLE]

if the (pointwise) limit exists. We call $\psi_{\mathcal{G}}(z)$ the “characteristic power series” of the graphon $\mathcal{G}$, and it is essentially a normalized limit of the reciprocal polynomials of the characteristic functions of $G$. In Theorem 3 below, we show that $\psi_{\mathcal{G}}$ is indeed well-defined, i.e., independent of the sequence of graphs left-converging to $\mathcal{G}$.

2 Characteristic Polynomials of Large Graphs

The following classical result will be useful in describing the coefficients of $\psi_{\mathcal{G}}(z)$. Let $\mathcal{H}_{k}$ – sometimes called “elementary graphs” – denote the family of unlabelled graphs on $k$ vertices each of whose components is an edge or a cycle.

Theorem 2** (Harary-Sachs [11]).**

Suppose $G$ is a graph on $n$ vertices, and $k\geq 0$ is an integer. The coefficient of $x^{n-k}$ in $\phi_{G}(x)$ is

[TABLE]

where $c(H)$ is the number of components of $H$, $z(H)$ is the number of cycles of $H$, and $[\#H\subseteq G]$ denotes the number of subsets of edges of $G$ which are isomorphic to $H$. (When $k=0$, $\mathcal{H}_{k}$ is the singleton consisting only of the empty graph $\epsilon$; take $c(\epsilon)=z(\epsilon)=0$.)

Clearly, if $H\in\mathcal{H}_{k}$, then $|E(H)|=k-c(H)+z(H)$. Also,

[TABLE]

where $H$ has $m_{i}$ components of size $a_{i}$ for each $i$. For simplicity, for a partition $\lambda$ where part $b_{i}$ occurs $m_{i}$ times for each $i$ (the $b_{i}$ all distinct), the quantity $\eta(\lambda)$ is defined by

[TABLE]

and we write $\eta(H)=\eta(\lambda(H))$ where $\lambda(H)$ is the integer partition of $|E(H)|$ given by the component cardinalities of $H$. If $\lambda$ is a partition of $n$, then the number of partitions of an $n$-set with structure $\lambda$ (a partition with $m_{i}$ parts of distinct sizes $b_{i}$) is given by

[TABLE]

Denote by $\Lambda^{\prime}_{n,k}$ the set of partitions of $n$ into $k$ parts, each of which is of size at least $2$; for $\lambda\in\Lambda^{\prime}_{n,k}$, denote its $i$-th largest part by $\lambda_{i}$, the $i$-th largest integer which appears as a part by $b_{i}$, and the multiplicity of $b_{i}$ by $m_{i}$.

We wish to transform the characteristic polynomial of a graph into a power series so that, as $n\rightarrow\infty$, the limit exists if $G_{n}$ converges to a graphon. This entails taking the reciprocal of $\phi_{G_{n}}(x)$, normalizing the coefficients with appropriate powers of $n$, and then letting $n$ tend to infinity. So, setting $z=n/x$, we have by Theorem 2,

[TABLE]

where $C_{2}$ denotes a single edge. Note that (treating this as a polynomial to avoid defining $0^{0}$), when $z=0$, the above expression equals $1$ because $\phi_{G}(x)$ is monic. Denote this polynomial by $\psi_{n}(z)$. Write $t(H,G)$ for the “homomorphism density” of the $k$-vertex graph $H$ in the $n$-vertex $G$, i.e., the number of (not necessarily injective) homomorphisms from $H$ to $G$ over $n^{k}$. Writing $\mu_{i}$ for the $i$-th eigenvalue of $A(G_{n})/n$, we may bound

[TABLE]

Then, since $t(C_{b},G_{n})=\sum_{i}\mu_{i}^{b}$ for each $b\geq 2$,

[TABLE]

since $\sum_{i}\mu_{i}=\operatorname{tr}A(G_{n})/n=0$. The above product converges if $\sum_{i}|\mu_{i}|$ does, which is $n^{-1}$ times the so-called “energy” $\mathcal{E}(G_{n})$ of $G_{n}$, the sum of its adjacency singular values. Since $\mathcal{E}(G_{n})$ can be as large as $Cn^{3/2}$, we should not hope for $\phi_{n}(z)$ always to converge. Indeed, $n^{3/2}$ tends to be the order of magnitude of the energy of dense graphs, i.e., graphs with $\Omega(n^{2})$ edges, the only graphs converging to a nontrivial graphon; see, for example, [14]. However, this is not always the case: indeed, $\mathcal{E}(K_{n})=2n-2$.

3 Characteristic Power Series of Graphons

In order to circumvent convergence problems discussed in the previous section, we introduce the so-called “regularized characteristic determinant” $\det^{(p)}(\mathcal{A})$ of a linear operator $\mathcal{A}$:

Definition 1**.**

For a positive integer $p$, the regularized characteristic determinant of a linear operator $\mathcal{A}$ with discrete spectrum is defined as

[TABLE]

where $\lambda_{j}$ varies over the eigenvalues of $\mathcal{A}$.

We then use this definition – albeit only the $p=2$ case, a.k.a. the Hilbert-Carleman determinant, after first considering $p=4$ – for reasons which will be apparent below, to define a characteristic power series of graphons:

Definition 2**.**

The characteristic power series of a graphon $\mathcal{G}$ is defined by

[TABLE]

By [9] (Chapter IV, Section 2), the function $\det^{(p)}(I-z\mathcal{A})$ is well-defined and entire (of genus $p-1$) for operators $\mathcal{A}$ in $\mathfrak{S}_{p}$, the operators which are Schatten $p$-class, i.e., for which the Schatten $p$-norm $(\sum_{i}\sigma_{i}^{p})^{1/p}$ is finite, where $\sigma_{i}$ are the singular values of $\mathcal{A}$, defined to be the eigenvalues of $\sqrt{\mathcal{A}^{\ast}\mathcal{A}}$. Since graphons give rise to self-adjoint operators, we will have throughout that $\sigma_{i}=|\lambda_{i}|$. Note that Schatten $2$-class bounded operators are the same as Hilbert-Schmidt operators, which all graphons’ corresponding integral transforms are; and Schatten $1$-class are the nuclear or trace-class operators, in which case $\det^{(1)}(I-z\mathcal{A})$ is the classical Fredholm determinant and $\operatorname{tr}(\mathcal{A})=\sum_{j}\lambda_{j}$ is the (signed) trace. It also follows from [9] (see Theorem IV.2.1) that $\det^{(p)}(I-z\mathcal{A})$ is continuous (uniform convergence on compact sets) with respect to convergence in $p$-norm of $\mathcal{A}$.

We now present our main theorem, demonstrating that $\psi_{\mathcal{G}}$ is indeed well-defined and is an entire function of Laguerre-Pólya class, i.e., a holomorphic function which is locally the limit of a series of polynomials whose roots are all real. Laguerre-Pólya functions have played a prominent role in the study of distributions of zeros of real polynomials and real entire functions (e.g., [2]), early 20th-century attempts to prove the Riemann hypothesis and a recent revival of such methods (see [10]), and classical complex analysis. The fact that $\psi_{\mathcal{G}}(z)$ is Laguerre-Pólya class implies that it has a Hadamard product expression (see, e.g., [12] p. 42–47):

[TABLE]

where $m$ is a nonnegative integer; $b$ and $c$ are real with $c\leq 0$; and $r$ ranges over the nonzero zeros of $\psi_{\mathcal{G}}(z)$. Note that the definition of $\psi_{\mathcal{G}}(z)$ is almost in this form already. In particular, $m=a=b=0$, and $c=(\|\mathcal{G}\|_{2}^{2}-\|\mathcal{G}\|_{1})/2$, and the product ranges over the reciprocals $r$ of the nonzero eigenvalues of $\mathcal{G}$.

Theorem 3**.**

Suppose the graphs $G_{n}$ converge to the graphon $\mathcal{G}$. For the sequence of functions $\psi_{n}(z)$ corresponding to the sequence of graphs $G_{n}$:

$\psi_{n}(z)$* converges pointwise as $n\rightarrow\infty$.* 2. 2.

$\psi_{n}(z)$* converges uniformly on compact sets as $n\rightarrow\infty$.* 3. 3.

Each coefficient of $\psi_{n}(z)$ converges as $n\rightarrow\infty$.

Furthermore, the limit is $\psi_{\mathcal{G}}(z)$, is entire of Laguerre-Pólya class, and its roots are the reciprocals of the nonzero eigenvalues of $\mathcal{G}$ (as a self-adjoint kernel operator) with multiplicity.

Proof.

Let $A^{\prime}_{n}=A(G_{n})/n$, so that

[TABLE]

There is a natural definition of the homomorphism densities $t(F,\mathcal{G})$ in terms of a certain integral which is standard for graphons, and [13] (7.22) shows that $\sum_{j}\lambda_{j}(\mathcal{G})^{k}=t(C_{k},\mathcal{G})$ for $k\geq 3$ and, by left-convergence, $t(C_{k},G_{n})\rightarrow t(C_{k},\mathcal{G})$ for $k\geq 3$. Thus, in particular, $\sum_{j}\lambda_{j}(\mathcal{G})^{4}$ converges, so $\mathcal{G}$ is Schatten $4$-class and $\det^{(4)}(I-z\mathcal{G})$ is entire. That $G_{n}\rightarrow\mathcal{G}$ implies that there exists a sequence of measure-preserving maps $\varphi_{n}$ of $[0,1]$ to itself so that, if $\mathcal{A}:[0,1]^{2}\rightarrow[0,1]$ is the kernel defined by $\mathcal{A}_{n}(x,y)=A(G_{n})_{\lceil xn\rceil,\lceil yn\rceil}$, then $\tilde{\mathcal{A}}_{n}(x,y):=\mathcal{A}_{n}(\varphi(x),\varphi(y))$ converges to $\mathcal{G}$ in cut norm; note that $\tilde{\mathcal{A}}_{n}$ has the same spectrum as $A^{\prime}_{n}$. By [13] Lemma 8.12, that $\tilde{\mathcal{A}}_{n}\rightarrow\mathcal{G}$ in cut norm implies that $\tilde{\mathcal{A}}_{n}\rightarrow\mathcal{G}$ in Schatten $4$-norm. As mentioned following Definition 2, Theorem IV.2.1 of [9] then implies that

[TABLE]

uniformly on compact sets. Since $\mathcal{G}$ is $\mathfrak{S}_{2}$-class, the function $\det^{(2)}(\mathcal{G})$ is defined and entire, so the right-hand side of (2) can be written

[TABLE]

Similarly, the left-hand side of (2) can be written

[TABLE]

because $\sum_{j}\lambda_{j}(A^{\prime}_{n})=\operatorname{tr}(A^{\prime}_{n})=0$. Since $t(C_{3},G_{n})\rightarrow t(C_{3},\mathcal{G})$, the cubic terms can be cancelled in (2) and the quadratic terms behave predictably, because $\sum_{j}\lambda_{j}(A)^{2}=\|\mathcal{G}\|_{2}^{2}$ (the Hilbert-Schmidt norm) and $\|G_{n}\|_{2}^{2}=2|E(G_{n})|/n^{2}$:

[TABLE]

Since the edge density $\sum_{j}\lambda_{j}(A^{\prime})^{2}=\|G_{n}\|_{2}^{2}=2|E(G_{n})|/n^{2}$ converges to $\|\mathcal{G}\|_{1}$, this can be rewritten as

[TABLE]

where the limit can be interpreted as uniform convergence on compact subsets of $\mathbb{C}$ or pointwise.

Then (1) and (2) follow, and, since (2) holds, Cauchy’s integral formula implies that (3) holds as well (by integrating $\psi_{n}(z)/(2\pi iz^{k})$ along a circle around the origin for each $k\geq 0$). Since $\psi_{n}(z)$ has only real roots (being the characteristic polynomial of a real symmetric matrix), the limit is of Laguerre-Pólya class. That the roots of $\psi_{\mathcal{G}}(z)$ are the reciprocals of the eigenvalues of the kernel operator corresponding to $\mathcal{G}$ now follows from Definitions 1 and 2. ∎

Theorem 3 has immediate consequences from various properties of determinants, for example the following result. Here we introduce the notation $\mathcal{G}\oplus_{p}\mathcal{H}$ for the $p$-disjoint union of $\mathcal{G}$ and $\mathcal{H}$, the graphon whose kernel $W$ is given by

[TABLE]

Corollary 1**.**

Given two graphons $\mathcal{G}$ and $\mathcal{H}$, the graphon $\mathcal{G}\oplus_{p}\mathcal{H}$ which is their disjoint union has the property that

[TABLE]

Proof.

The kernel of $\mathcal{G}\oplus_{p}\mathcal{H}$ is $\mathcal{G}^{\prime}+\mathcal{H}^{\prime}$, where $\mathcal{G}^{\prime}(x,y)=\mathcal{G}(x/p,y/p)$ and $\mathcal{H}^{\prime}(x,y)=\mathcal{H}((x-p)/(1-p),(y-p)/(1-p))$ (interpreting functions to be zero outside $[0,1]\times[0,1]$). By [9] (Section VI.2) and the fact that $\mathcal{G}^{\prime}\mathcal{H}^{\prime}=0$ (multiplication interpreted as composition), the Hilbert-Carleman determinant satisfies

[TABLE]

But $\mathcal{G}^{\prime}\mathcal{H}^{\prime}=0$, so $\sum_{i}\lambda_{i}(\mathcal{G}^{\prime}\mathcal{H}^{\prime})=0$. Then

[TABLE]

∎

One can also obtain the above result by viewing $\mathcal{G}^{\prime}\oplus_{p}\mathcal{H}^{\prime}$ as an operator on a direct sum of Hilbert spaces, whose spectrum is easy to describe.

4 Special Cases

Recall that $\operatorname{tr}\mathcal{G}=\sum_{j}\lambda_{j}$, the sum of the eigenvalues of (the kernel of) $\mathcal{G}$, and that $\mathcal{G}$ is “trace class” (aka “nuclear”) if this sum converges absolutely. When $\mathcal{G}$ is trace class, we may write

[TABLE]

a factorization of the characteristic power series into a monic polynomial-like product whose roots are $\{1/s:s\in\mathcal{S}\}$ where $\mathcal{S}$ is the set of nonzero eigenvalues of $\mathcal{G}$, and an exponential term. The quantity $\operatorname{tr}\mathcal{G}$ is zero if $\mathcal{G}$ is bipartite: in particular, the eigenvalues comes in pairs $\pm\lambda_{j}$. (For more on the spectra of bipartite graphs, see [8], in particular Theorem 8.) In this case, $\psi_{G}(z)$ has no monomials of odd degree:

[TABLE]

Furthermore, $\|\mathcal{G}\|_{2}^{2}=\|\mathcal{G}\|_{1}^{1}$ iff $\mathcal{G}$ is a [math]-$1$ function except for a set of measure zero, as with a simple blow-up of a graph (sometimes called a “pixel diagram”), so the quadratic term vanishes in the exponential, resulting in

[TABLE]

We can also use (3) to obtain a simple expression for the characteristic power series of $p$-quasi-random graphons.

Proposition 1**.**

A sequence of graphs $G_{n}$ is $p$-quasirandom iff it converges to a graphon $\mathcal{G}$ with

[TABLE]

Furthermore, when this occurs, we can write

[TABLE]

where $\Lambda(k;i,j)$ is the set of integer partitions of $k$ into $j$ parts of size at least $2$, of which $i$ are of size exactly $2$.

Proof.

It is clear from Theorem 1 that, if $G_{n}$ is a $p$-quasirandom sequence, then it converges to a graphon $\mathcal{G}$ which has only one nonzero eigenvalue $\lambda=p$, and furthermore $(\|\mathcal{G}\|_{2}^{2}-\|\mathcal{G}\|_{1})/2=-p(1-p)/2$, so

[TABLE]

On the other hand, if (6) holds, then $\mathcal{G}$ has only one non-zero eigenvalue, and $\psi_{\mathcal{G}}$ has only one root, at $z=1/p$. By the Spectral Theorem, this also holds if and only if the graphon $\mathcal{G}$, regarded as a function of $[0,1]^{2}$ up to modification on a set of measure zero, can be written as $p\phi(x)\phi(y)$ for some function $\phi$ normalized so that $\|\phi\|_{2}=1$. Since the quadratic term in expression (6) is $-p(1-p)/2=(\|\mathcal{G}\|_{2}^{2}-\|\mathcal{G}\|_{1})/2$, and $\|\mathcal{G}\|_{2}^{2}=p^{2}$, this implies that $\|\mathcal{G}\|_{1}=p$. But then

[TABLE]

so that $\|\phi\|_{1}=\|\phi\|_{2}$, $\phi$ is the constant function, and $\mathcal{G}$ is $p$-quasirandom.

To see that $\psi_{\mathcal{G}}$ can also be written as in (5), observe that, by Theorem 1 and 3, as well as the discussion following Theorem 2,

[TABLE]

because partitions of $k$ into parts of size at least $2$ correspond bijectively to elements of $\mathcal{H}_{k}$. ∎

It is also straightforward to see directly that the middle and right-hand expressions in 2 are equal. If $\alpha(z)=\exp(\beta(z))$ is a power series, where the $z^{n}$ coefficient of $\alpha$ is $a_{k}$ and the $z^{k}$ coefficient of $\beta$ is $b_{k}$, then (by standard facts about exponential generating functions, see, e.g., [3]), letting $\lambda$ be an integer partition with $m_{i}$ parts of distinct sizes $c_{i}$, $i=1$ to $t$,

[TABLE]

where $\Pi$ is the set of (set) partitions of an $k$-set. Thus, if

[TABLE]

and $\alpha(z)=\exp(\beta)$, then

[TABLE]

5 Questions

It is tempting to define instead a characteristic power series without the regularization, i.e., $\det(I-z\mathcal{G})=\prod_{j}(1-\lambda_{j}z)$, the “Fredholm determinant”. However, this product may not converge absolutely. Using Fourier series, it is straightforward to show that, for the graphon $\mathcal{G}(x,y)$ defined by

[TABLE]

we have

[TABLE]

Since $\{f_{n}\}_{n=0}^{\infty}$ with $f_{n}(t)=\sqrt{2}\cos[(2n+1)\pi t/2]\}$ is an orthonormal family of functions on $[0,1]$, this shows that the spectrum of $\mathcal{G}$ is $\left\{\frac{2(-1)^{n}}{(2n+1)\pi}\right\}_{n=0}^{\infty}$. Since the (odd) harmonic series diverges, it follows that $\mathcal{G}$ is not trace-class, i.e., $\sum_{i}\lambda_{i}$ does not converge absolutely. However, it still converges conditionally, so $\det(I-z\mathcal{G})$ is well-defined.

One can also construct a function from $[0,1]\times[0,1]\rightarrow\mathbb{R}^{2}$ by setting $\mathcal{G}(x,y)=\sum_{n=0}^{\infty}\frac{1}{n}(-1)^{\epsilon_{n}(x)+\epsilon_{n}(y)}$ where $\epsilon_{n}(x)$ is the $n$-th digit of $x$ in binary. Since $\{(-1)^{\epsilon_{n}}\}_{n=0}^{\infty}$ is the orthonormal Rademacher system, the spectrum of $\mathcal{G}$ is the harmonic series. Thus $\det(I-z\mathcal{G})$ is undefined, but because random harmonic series have full support on the real line, the function $\mathcal{G}$ is not a graphon (and cannot be linearly scaled to become one).

Thus, we are led to the following question.

Question 1**.**

Does there exist a graphon $\mathcal{G}$ for which the Fredholm determinant $\det(I-z\mathcal{G})$ does not exist?

The characteristic polynomial of the $p$-quasirandom graphon has some possible connections with its probabilistic interpretations, as follows.

Question 2**.**

Let $\mathcal{G}$ be the $p$-quasirandom graphon. The function

[TABLE]

has some unexplained connections with Gaussian probability distributions. If $M(z)$ is the moment generating function of a normal distribution of mean $p$ and variance $p(1-p)$ – the normalized limit of a [math]-$1$ random walk with bias $p$ – then $\psi_{\mathcal{G}}(z)=(1-pz)/M(-z)$. Why?

Our next question concerns to what extent some of the above approach can be applied to the many other well-known graph polynomials: matching polynomial, Laplacian characteristic polynomial, chromatic polynomial, etc. When is it the case that, given some notion of graph convergence, such as left-convergence leading to graphons as above or Benjamini-Schramm convergence of very sparse graphs, these polynomials when suitably normalized converge to some power series? One motivation for asking this is a related, growing body of work on limits of measures supported on the roots of natural graph polynomials. For example, building off of work by Sokal [16] and Borgs-Chayes-Kahn-Lovász [4], Abért-Hubai [1] showed the convergence of harmonic moments (quantities $\int_{K}f\,d\nu$ for holomorphic functions $f$ and certain regions $K\subset\mathbb{C}$) of the uniform probability distribution over the chromatic roots of Benjamini-Schramm-convergence graph sequences; subsequently, Csikvári-Frenkel [6] generalized this to a wide class of graph polynomials (including the characteristic polynomial) and Csikvári-Frenkel-Hladký-Hubai [7] showed that it holds even for dense graph (i.e., graphon) convergence with suitable normalization. From a different perspective, [18] empirically showed that chromatic roots of Erdős-Rényi random graphs appear to have a scaling limit.

Question 3**.**

For which other graph polynomials and graph limit process can the above type of analysis be carried out? How about for hypergraphs?

Finally, we mention a question that arose in the context of experimentally computing the coefficients of the characteristic power series of that simplest of graphons, the uniform quasirandom graphon.

Question 4**.**

It is straightforward to show (by, for example, applying Turán’s Inequalities; see [15]) that the coefficients $c_{k}$ of $\psi_{\mathcal{G}}$ are log-concave for any graphon $\mathcal{G}$, but the consequences of this for unimodality are unclear because we do not know the sign pattern of the coefficients of $\psi_{\mathcal{G}}(z)$. For example, the characteristic power series of a $p=1/2$ quasi-random graphon has sign pattern

[TABLE]

More specifically, can $c_{k}=0$ if $k\neq 1$?

6 Acknowledgments

Thanks to G. Clark, C. Edgar, G. Fickes, V. Nikiforov, and A. Riasanovsky for helpful discussions. Thanks also to the anonymous referees for their patience and very useful suggestions, especially the identification of a need to use higher-order determinants.