# Eigenvalue bounds for some classes of matrices associated with graphs

## TL;DR

This paper introduces new eigenvalue bounds for matrices related to graphs, improving understanding of spectral properties of regular graphs, normalized adjacency, and Laplacian matrices with verified sharpness.

## Contribution

It presents two novel eigenvalue inclusion sets and derives bounds for key eigenvalues of various graph-associated matrices, enhancing spectral analysis tools.

## Key findings

- New eigenvalue bounds for adjacency matrices of regular graphs
- Bounds for eigenvalues of normalized adjacency matrices
- Bounds for Laplacian eigenvalues with verified sharpness

## Abstract

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular graphs. Then, we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest eigenvalue and the largest eigenvalue of the Laplacian matrices of graphs. Sharpness of these bounds are verified by examples.