Maximization of the spectral radius of block graphs with a given dissociation number

Abstract

A connected graph is called a block graph if each of its blocks is a complete graph. Let $\mathbf{Bl}(\textbf{k}, \varphi)$ be the class of block graphs on $\textbf{k}$ vertices with given dissociation number $\varphi$. In this article, we have shown the existence and uniqueness of a block graph $\mathbb{B}_{\textbf{k},\varphi}$ in $\mathbf{Bl}(\textbf{k}, \varphi)$ that maximizes the spectral radius $\rho(G)$ among all graphs $G$ in $\mathbf{Bl}(\textbf{k}, \varphi)$. Furthermore, we also provide bounds on $\rho(\mathbb{B}_{\textbf{k},\varphi})$.