Minimal spectral radius of graphs with given matching number

The Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. This paper addresses the Brualdi-Solheid problem for \( \mathcal{G}_{n,β} \), the family of graphs with order \( n \) and matching number \( β\), aiming to identify its spectrally minimal graphs i.e., those that minimize the spectral radius \(ρ(G)\). We introduce the novel concept of ``quasi-adjacency'' relation, developing a unified structural classification framework for trees in \(\mathcal{G}_{n,β}\), which clarifies structural properties and provides a constructive method to generate trees with fixed \(β\). By showing that all spectrally minimal graphs in \( \mathcal{G}_{n,β} \) are trees, we further narrow the search for extremal graphs. Additionally, we apply this framework to the representative cases \(β=2,3,4\), obtaining the minimizers by explicit structural formulas involving parameters related to \(n\).