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 \( Gn,β \), 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 \((G)\). We introduce the novel concept of ``quasi-adjacency'' relation, developing a unified structural classification framework for trees in \(Gn,β\), which clarifies structural properties and provides a constructive method to generate trees with fixed \(β\). By showing that all spectrally minimal graphs in \( Gn,β \) 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 \(n\).
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