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Minimal spectral radius of graphs with given matching number

Jiaqi Liu, Zhenzhen Lou, Vilmar Trevisan

TL;DR

This work resolves the Brualdi-Hoffman problem for graphs with fixed matching number by proving that spectrally minimal graphs in $\mathcal{G}_{n,\beta}$ are necessarily trees and by developing a unifying quasi-adjacency framework to classify such trees. It shows that a size-$\beta$ dominating set with controlled quasi-adjacency properties governs the minimizers and then derives explicit minimizers for $\beta=2,3,4$ using constructive parametrizations and existing results. The paper leverages spectral comparison techniques, spanning-tree preservation of the matching number, and Perron-vector-based comparisons to exclude non-minimizers. The resulting explicit minimizers provide concrete, case-based formulas, advancing the understanding of extremal spectral graph theory under a fixed matching constraint and offering precise templates for constructing extremal trees.

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 \(n\).

Minimal spectral radius of graphs with given matching number

TL;DR

This work resolves the Brualdi-Hoffman problem for graphs with fixed matching number by proving that spectrally minimal graphs in are necessarily trees and by developing a unifying quasi-adjacency framework to classify such trees. It shows that a size- dominating set with controlled quasi-adjacency properties governs the minimizers and then derives explicit minimizers for using constructive parametrizations and existing results. The paper leverages spectral comparison techniques, spanning-tree preservation of the matching number, and Perron-vector-based comparisons to exclude non-minimizers. The resulting explicit minimizers provide concrete, case-based formulas, advancing the understanding of extremal spectral graph theory under a fixed matching constraint and offering precise templates for constructing extremal trees.

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 , the family of graphs with order and matching number , 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 , which clarifies structural properties and provides a constructive method to generate trees with fixed . By showing that all spectrally minimal graphs in are trees, we further narrow the search for extremal graphs. Additionally, we apply this framework to the representative cases , obtaining the minimizers by explicit structural formulas involving parameters related to .
Paper Structure (5 sections, 28 theorems, 19 equations, 12 figures)

This paper contains 5 sections, 28 theorems, 19 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Quasi-adjacent graphs
  3. The characteristic of minimizers
  4. The minimizers for $\beta <4$
  5. Proof of Theorem \ref{['thm-1.3']}

Key Result

Theorem 1.1

Let $T_2^*$ be the graph with the minimal spectral radius among all graphs in $\mathcal{G}_{n,2}$, with $n\geq 4$. Then $T_2^* \cong T_2\left( \left\lfloor \tfrac{n-3}{2} \right\rfloor, \left\lceil \tfrac{n-3}{2} \right\rceil \right)$(see Fig. fig-2, left).

Figures (12)

  • Figure 1: The graphs $T_{2}(a,b)$ (left) and $T_{3}(a,b,c)$ (right) .
  • Figure 2: The graphs $T_{6}(a,b,c,d)$ (left) and $T_{10}(a,b,c,d)$ (right) .
  • Figure 3: A graph $T$ (left) and the quasi-adjacent graph of $T$ with respect to $X=\{v_1, v_3, v_4\}$.
  • Figure 4: Five possible structures.
  • Figure 5: A partition of $X$.
  • ...and 7 more figures

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • ...and 52 more