On the spectral radius of graphs: nonregular distance-hereditary graphs with given edge-connectivity, graphs with tree-width $k$ and block graphs with prescribed independence number $\alpha$

TL;DR

This paper establishes bounds on the spectral radius of certain graph classes, relating it to edge connectivity, and identifies unique extremal graphs within these classes.

Contribution

It provides new lower bounds for the spectral radius in nonregular distance-hereditary graphs based on edge connectivity and characterizes extremal graphs with maximum spectral radius.

Findings

Lower bound for Δ(G)-ρ(G) in nonregular distance-hereditary graphs

Unique graph maximizes spectral radius in specified classes

Spectral radius reaches maximum at a unique graph in the class

Abstract

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph $G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present a lower bound for $\Delta(G)-\rho(G)$ in terms of the edge connectivity of $G$, where $G$ is a nonregular distance-hereditary graph. We also prove that $\rho(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded tree-width or is in the class of block graphs with prescribed independence number.