On the spectral radius of clique trees with a given zero forcing number

TL;DR

This paper investigates the maximum spectral radius of clique trees with a fixed number of vertices and zero forcing number, establishing the existence, uniqueness, and bounds for extremal graphs within this class.

Contribution

It proves the existence and uniqueness of the extremal clique tree with maximal spectral radius in the class G(n,k), and provides an upper bound for this spectral radius.

Findings

Existence of a unique extremal clique tree with maximum spectral radius.

An upper bound for the spectral radius of the extremal graph.

Characterization of clique trees with given zero forcing number.

Abstract

Let $G(n,k)$ be the class of clique trees on $n$ vertices and zero forcing number $k$, where $\left \lfloor \frac{n}{2} \right \rfloor + 1 \le k \le n-1$ and each block is a clique of size at least $3$. In this article, we proved the existence and uniqueness of a clique tree in $G(n,k)$ that attains maximal spectral radius among all graphs in $G(n,k)$. We also provide an upper bound for the spectral radius of the extremal graph.