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

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.