Lower bounds for the isoperimetric numbers of random regular graphs

TL;DR

The paper addresses lower bounds for vertex and edge isoperimetric numbers in random $d$-regular graphs, focusing on the vertex variant at $u=1/2$ and deriving asymptotic results as $d\to\infty$. It employs the pairing model and a first-moment method to count subsets with specified boundary sizes, introducing the vertex-expansion number $I_{V,u}(d)$ and the threshold $H_d(u)$, then derives explicit lower bounds $i_{V,u}(d)\ge H_d(u)/u$ and, for $u=1/2$, $i_V(d)\ge A_d(1/2)$ with $A_d(1/2)=1-2/d+O((\log d)/d^2)$. Parallel arguments yield bounds for the edge version, defining $\widehat{A}_d(u)$ and showing $i_{E,u}(d)\ge \widehat{A}_d(u)$ with $i_{E,u}(d)\ge d(1-u+o(1))$ as $d\to\infty$. The results sharpen our understanding of expansion properties in random regular graphs and provide explicit, verifiable bounds that complement spectral approaches.

Abstract

The vertex isoperimetric number of a graph $G=(V,E)$ is the minimum of the ratio $|\partial_{V}U|/|U|$ where $U$ ranges over all nonempty subsets of $V$ with $|U|/|V|\le u$ and $\partial_{V}U$ is the set of all vertices adjacent to $U$ but not in $U$. The analogously defined edge isoperimetric number---with $\partial_{V}U$ replaced by $\partial_{E}U$, the set of all edges with exactly one endpoint in $U$---has been studied extensively. Here we study random regular graphs. For the case $u=1/2$, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all $d\ge3$. Moreover, we obtain a lower bound on the asymptotics as $d\to\infty$. We also provide asymptotically almost sure lower bounds on $|\partial_{E}U|/|U|$ in terms of an upper bound on the size of $U$ and analyse the bounds as $d\to\infty$.

Theorems & Definitions (22)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof