Minimal spectral radius of graphs with given matching number

TL;DR

This paper determines the graphs with the smallest spectral radius among all graphs with a fixed number of vertices and matching number, introducing a new structural classification framework for trees.

Contribution

It introduces the concept of quasi-adjacency and develops a unified framework to classify and generate extremal trees with given matching number.

Findings

Spectrally minimal graphs are trees within the class.

Explicit structural formulas for minimizers when eta=2,3,4.

All extremal graphs have minimal spectral radius in their class.

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 \( \mathcal{G}_{n,\beta} \), 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 \(n\).