Minimal spectral radius of graphs with given matching number

Abstract: 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).