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

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 $\gamma (G)$. 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 ${\mathcal{U}}_{n,\gamma }$ of simple connected unicyclic graphs on $n$ vertices with matching number $\gamma$. We specifically characterize all graphs that achieve the minimum spectral radius in ${\mathcal{U}}_{n,\gamma }$ for matching numbers $\gamma \in \{1,2,3,\lfloor \frac{n}{2}\rfloor \}$.