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