A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

TL;DR

This paper establishes a lower bound for the smallest eigenvalue of regular graphs with many small subgraphs, generalizing previous results, and applies it to analyze the eigenvalues of the associahedron graph.

Contribution

It provides a new lower bound for the smallest eigenvalue in regular graphs containing multiple copies of a fixed subgraph, extending prior work on triangles.

Findings

Lower bound for smallest eigenvalue of regular graphs with subgraphs

Application of bound to associahedron graph's eigenvalues

Order of magnitude of the associahedron graph's smallest eigenvalue

Abstract

In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph.