$X$-Ramanujan Graphs

TL;DR

This paper introduces a new class of infinite graphs called additive product graphs and demonstrates the existence of infinitely many finite graphs with spectral properties similar to their infinite counterparts, extending Ramanujan graph theory.

Contribution

It develops the concept of additive product graphs and the additive characteristic polynomial, generalizing known polynomials and constructions, and proves the existence of $k$-quasi-$X$-Ramanujan graphs for these structures.

Findings

Existence of infinitely many $k$-quasi-$X$-Ramanujan graphs for additive product graphs.

Introduction of the additive characteristic polynomial and proof of its real-rootedness.

Generalization of Ramanujan graph properties beyond trees to additive product graphs.

Abstract

Let $X$ be an infinite graph of bounded degree; e.g., the Cayley graph of a free product of finite groups. If $G$ is a finite graph covered by $X$, it is said to be $X$-Ramanujan if its second-largest eigenvalue $\lambda_2(G)$ is at most the spectral radius $\rho(X)$ of $X$, and more generally $k$-quasi-$X$-Ramanujan if $\lambda_k(G)$ is at most $\rho(X)$. In case $X$ is the infinite $\Delta$-regular tree, this reduces to the well known notion of a finite $\Delta$-regular graph being Ramanujan. Inspired by the Interlacing Polynomials method of Marcus, Spielman, and Srivastava, we show the existence of infinitely many $k$-quasi-$X$-Ramanujan graphs for a variety of infinite $X$. In particular, $X$ need not be a tree; our analysis is applicable whenever $X$ is what we call an additive product graph. This additive product is a new construction of an infinite graph $\mathsf{AddProd}(A_1, \dots, A_c)$ from finite 'atom' graphs $A_1, \dots, A_c$ over a common vertex set. It generalizes the notion of the free product graph $A_1 * \cdots * A_c$ when the atoms $A_j$ are vertex-transitive, and it generalizes the notion of the universal covering tree when the atoms $A_j$ are single-edge graphs. Key to our analysis is a new graph polynomial $\alpha(A_1, \dots, A_c;x)$ that we call the additive characteristic polynomial. It generalizes the well known matching polynomial $\mu(G;x)$ in case the atoms $A_j$ are the single edges of $G$, and it generalizes the $r$-characteristic polynomial introduced in [Ravichandran'16, Leake-Ravichandran'18]. We show that $\alpha(A_1, \dots, A_c;x)$ is real-rooted, and all of its roots have magnitude at most $\rho(\mathsf{AddProd}(A_1, \dots, A_c))$. This last fact is proven by generalizing Godsil's notion of treelike walks on a graph $G$ to a notion of freelike walks on a collection of atoms $A_1, \dots, A_c$.

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