On the distance $\alpha$-spectral radius of a connected graph

TL;DR

This paper investigates the distance alpha-spectral radius of connected graphs, providing bounds, transformation techniques, and characterizations of extremal graphs for this spectral measure.

Contribution

It introduces bounds for the distance alpha-spectral radius, proposes graft transformations affecting this radius, and characterizes extremal graphs within certain classes.

Findings

Bounds established for the distance alpha-spectral radius.

Graft transformations that modify the spectral radius.

Identification of graphs with extremal spectral radius values.

Abstract

For a connected graph $G$ and $\alpha\in [0,1)$, the distance $\alpha$-spectral radius of $G$ is the spectral radius of the matrix $D_{\alpha}(G)$ defined as $D_{\alpha}(G)=\alpha T(G)+(1-\alpha)D(G)$, where $T(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. We give bounds for the distance $\alpha$-spectral radius, especially for graphs that are not transmission regular, propose some graft transformations that decrease or increase the distance $\alpha$-spectral radius, and determine the unique graphs with minimum and maximum distance $\alpha$-spectral radius among some classes of graphs.