On the spectral radius of block graphs having all their blocks of the same size

TL;DR

This paper investigates the spectral radius of a specific class of block graphs with uniform block size, establishing conditions for minimal spectral radius and providing bounds.

Contribution

It characterizes the unique graph with minimal spectral radius in the class and offers a lower bound for the spectral radius of such graphs.

Findings

Unique graph attains minimum spectral radius under given conditions.

Provides a lower bound for the spectral radius in the class.

Analyzes spectral properties of block graphs with uniform blocks.

Abstract

Let $\mathcal{B}(n,q)$ be the class of block graphs on $n$ vertices having all their blocks of the same size. We prove that if $G\in \mathcal{B}(n,q)$ has at most three pairwise adjacent cut vertices then the minimum spectral radius $\rho(G)$ is attained at a unique graph. In addition, we present a lower bound for $\rho(G)$ when $G\in \mathcal{B}(n,q)$.