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The spectral radii and extremal graphs of two types of minimal graphs

Liwen Lian, Jinfeng Liu, Mengyuan Niu, Xiumei Wang

TL;DR

This work addresses sharp spectral-radius bounds for two classes of minimal graphs: minimal matching-covered bipartite graphs and minimal factor-critical graphs. It employs ear decomposition and edge-exchange techniques to derive exact, order-dependent bounds and to identify unique extremal graphs: $P^{*}_{3}$ for the bipartite case and $K_{1} \vee \frac{n-1}{2}K_{2}$ for the factor-critical case, with small-$n$ cases aligning with cycles. The results solve a Brualdi–Solheid-type extremal problem for these graph families and highlight the role of ear compatibility and chordless cycles in spectral extremality. Overall, the paper provides complete characterizations of extremal graphs and explicit spectral radii, enriching the theory of spectral extremal graph problems for structured minimal graphs.

Abstract

A connected nontrivial graph $G$ is {\it matching covered} if every edge of $G$ is contained in some perfect matching of $G$. A matching covered graph $G$ is {\it minimal} if $G-e$ is not matching covered for each edge $e$ of $G$. A graph is said to be {\it factor-critical} if $G-v$ has a perfect matching for every $v\in V(G)$. A factor-critical graph $G$ is said to be {\it minimal factor-critical} if $G-e$ is not factor-critical graph for each edge $e\in E(G)$. In this paper, by employing ear decomposition and edge-exchange techniques, the greatest spectral radii of minimal matching covered bipartite graphs and minimal factor-critical graphs are determined, and the corresponding extremal graphs are characterized.

The spectral radii and extremal graphs of two types of minimal graphs

TL;DR

This work addresses sharp spectral-radius bounds for two classes of minimal graphs: minimal matching-covered bipartite graphs and minimal factor-critical graphs. It employs ear decomposition and edge-exchange techniques to derive exact, order-dependent bounds and to identify unique extremal graphs: for the bipartite case and for the factor-critical case, with small- cases aligning with cycles. The results solve a Brualdi–Solheid-type extremal problem for these graph families and highlight the role of ear compatibility and chordless cycles in spectral extremality. Overall, the paper provides complete characterizations of extremal graphs and explicit spectral radii, enriching the theory of spectral extremal graph problems for structured minimal graphs.

Abstract

A connected nontrivial graph is {\it matching covered} if every edge of is contained in some perfect matching of . A matching covered graph is {\it minimal} if is not matching covered for each edge of . A graph is said to be {\it factor-critical} if has a perfect matching for every . A factor-critical graph is said to be {\it minimal factor-critical} if is not factor-critical graph for each edge . In this paper, by employing ear decomposition and edge-exchange techniques, the greatest spectral radii of minimal matching covered bipartite graphs and minimal factor-critical graphs are determined, and the corresponding extremal graphs are characterized.

Paper Structure

This paper contains 4 sections, 20 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal matching covered bipartite graphs
  4. Minimal factor-critical graphs

Key Result

Theorem 1.1

Let $G$ be a minimal matching covered bipartite graph of order $n$. If $n\leq4$, then $\rho(G)\leq2$, the equality holds if and only if $G\cong C_{4}$$\mathrm{;}$ If $n\geq 6$, then $\rho(G)\leq\frac{1+\sqrt{2n-3}}{2}$, the equality holds if and only if $G\cong P^{*}_{3}$, where $P^{*}_{3}$ is the u

Figures (3)

  • Figure 1: The graph $P^{*}_{3}$.
  • Figure 2: $P_{k}$, $\mathcal{P}_{k+1}$ and $\mathcal{P}_{k+2}$
  • Figure 3: the local structures of $P^{m-1}$, $\mathcal{P}^{m}_{1}$ and $\mathcal{P}^{m}_{2}$

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: ref16
  • Lemma 2.4: ref16
  • Lemma 2.5: ref16
  • Lemma 2.6: ref16
  • Lemma 2.7: ref18
  • Lemma 2.8: ref200
  • ...and 20 more