Spectral extremal problems for $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs

TL;DR

This paper investigates spectral radius conditions that guarantee the existence of $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs, generalizing previous factor theorems and resolving open problems for $k=0$.

Contribution

It introduces spectral conditions for $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs, extending factor theory and solving open problems when $k=0$.

Findings

Established spectral radius criteria for $(a,b,k)$-critical graphs.

Provided spectral conditions that resolve open problems for $k=0$.

Generalized factor theorems to fractional and critical cases.

Abstract

A factor of a graph is essentially a specific type spanning subgraph. The study of characterizing the existence of $[a, b]$-factors based on eigenvalue conditions can be traced back to the work of Brouwer and Haemers (2005) on perfect matchings. With the advancement of graphs factor theory, the related spectral extremal problems, particularly the study of $[a,b]$-factors and fractional $[a,b]$-factors, have been widely studied by scholars. Our work is motivated by research related to the $[a,b]$-factors and fractional $[a,b]$-factors, and explores their generalizations: $(a,b,k)$-critical graphs and fractional $(a,b,k)$-critical graphs. A graph $G$ is called an $(a,b,k)$-critical (a fractional $(a,b,k)$-critical) graph if after deleting any $k$ vertices of $G$ the remaining graph of $G$ has an $[a,b]$-factor (a fractional $[a,b]$-factor). In this paper, we establish spectral radius conditions for a graph to be $(a,b,k)$-critical or fractional $(a,b,k)$-critical. When $k=0$, our results also resolve some open problems concerning $[a, b]$-factors and fractional $[a, b]$-factors.