Tight products and Expansion

TL;DR

This paper introduces the concept of tight products of graphs, explores conditions for their existence, and presents new models for random regular graphs with bounded second eigenvalues, advancing graph theory and spectral analysis.

Contribution

It defines tight products of graphs, characterizes when they exist, and develops a new random graph model with spectral properties similar to lifts but using fewer random bits.

Findings

Characterization of when tight products exist for certain graphs

New model of random d-regular graphs with eigenvalues at most O(d^{3/4})

Connection to class-1 (2k+1)-regular graphs

Abstract

In this paper we study a new product of graphs called {\em tight product}. A graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$ and $G_2$, if $V(H)=V(G_1)\times V(G_2)$ and both projection maps $V(H)\to V(G_1)$ and $V(H)\to V(G_2)$ are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is $NP$-hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class-1 $(2k+1)$-regular graphs. We also obtain a new model of random $d$-regular graphs whose second eigenvalue is almost surely at most $O(d^{3/4})$. This construction resembles random graph lifts, but requires fewer random bits.