# Bounds on the $\alpha$-distance spectrum of graphs

## TL;DR

This paper introduces bounds on the spectral radius, energy, and Estrada index related to the $ ext{alpha}$-distance matrix of graphs, expanding spectral graph theory with new spectral bounds and indices.

## Contribution

It defines the $ ext{alpha}$-distance Estrada index and provides bounds on its spectral radius, energy, and Estrada index, advancing the understanding of spectral properties of $ ext{alpha}$-distance matrices.

## Key findings

- Bounds on the spectral radius of $D_{\alpha}(G)$
- Bounds on the $\alpha$-distance energy of $G$
- Bounds on the $\alpha$-distance Estrada index

## Abstract

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $\alpha$-distance energy of $G$ was defined based on the spectra of $D_{\alpha}(G)$. In this paper, we define the $\alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{\alpha}(G)$. And we give some bounds on the spectral radius of $D_{\alpha}(G)$, $\alpha$-distance energy and $\alpha$-distance Estrada index of $G$.