Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues

TL;DR

This paper investigates the minimum graph size ensuring a certain number of nonpositive eigenvalues in the adjacency matrix, establishing new bounds for Laplacian eigenvalues and connecting graph eigenvalue properties with Ramsey and triangular numbers.

Contribution

It introduces the function NPO(k), determines its values for small k, and links it to Ramsey and triangular numbers to derive new lower bounds for Laplacian eigenvalues.

Findings

NPO(k) values are 1, 3, 6, 10, 16 for k=1 to 5.

For all k ≥ 5, NPO(k) exceeds T_k and is at least R(k,k+1).

The k-th Laplacian eigenvalue is bounded below by the NPO(k)-th degree.

Abstract

Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any undirected graph with $n$ vertices or more has at least $k$ nonpositive eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$ is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree, which generalizes some prior results.