Extrema of graph eigenvalues

TL;DR

This paper derives tight bounds for the eigenvalues and singular values of graphs, introduces new constructions for open spectral problems, and explores related matrix problems, advancing understanding of graph spectra and their extremal properties.

Contribution

The paper provides the first tight bounds for the $k$'th largest eigenvalue of graphs for all $k>2$, and introduces novel constructions using strongly regular and Hadamard matrices for spectral extremal problems.

Findings

Derived tight bounds for all $k>2$ eigenvalues of graphs.

Constructed graphs with extremal spectral properties using Hadamard matrices.

Extended spectral bounds and problems to $(-1,1)$-matrices.

Abstract

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th largest singular value $\lambda_{k}^{\ast}(G).$ Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can $\lambda_{k}(G)+\lambda_{k}(\bar{G})$ be$?$ These constructions are successful also in another open question: How large can the Ky Fan norm $\lambda_{1}^{\ast}(G)+...+\lambda_{k}^{\ast }(G)$ be $?$ Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for $(-1,1)$-matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study.