On the minimum spectral radius of unicyclic graphs with a given matching number

Abstract

A matching $M$ in a graph $G = (V, E)$ is a set of edges such that no two edges in $M$ share a common vertex. A matching with maximum cardinality is called a maximum matching and its cardinality is the matching number $γ(G)$. The spectral radius of $G$ is the maximum absolute eigenvalue of its adjacency matrix. This article addresses the Brualdi-Solheid problem--the determination of extremal spectral radii within specific graph classes--for the class $\mathcal{U}_{n,γ}$ of simple connected unicyclic graphs on $n$ vertices with matching number $γ$. We specifically characterize all graphs that achieve the minimum spectral radius in $\mathcal{U}_{n,γ}$ for matching numbers $γ\in \left\{ 1, 2, 3, \lfloor \frac{n}{2} \rfloor \right\}$.