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

TL;DR

This paper identifies the unique block graph with a given number of vertices and dissociation number that maximizes spectral radius, providing existence, uniqueness, and bounds for this extremal graph.

Contribution

It establishes the existence and uniqueness of the spectral radius-maximizing block graph with specified parameters and derives bounds for its spectral radius.

Findings

Existence and uniqueness of the extremal block graph.

Bounds on the spectral radius of the extremal graph.

Characterization of the maximizing graph within the class.

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 $\rho(\mathbb{B}_{\textbf{k},\varphi})$.