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

TL;DR

This paper determines the maximum spectral radii of minimal matching covered bipartite graphs and minimal factor-critical graphs, characterizing the extremal graphs using ear decomposition and edge-exchange methods.

Contribution

It introduces new results on spectral radii bounds for specific minimal graph classes and characterizes the extremal graphs within these classes.

Findings

Identified the greatest spectral radii for minimal matching covered bipartite graphs.

Identified the greatest spectral radii for minimal factor-critical graphs.

Characterized the extremal graphs achieving these spectral radii.

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.