Spectral extremal problems for fractional $ID$-$[a,b]$-factor-critical graphs

Abstract

A factor of a graph is essentially a specific type spanning subgraph. In recent years, the spectral extremal problem of characterizing the existence of graph factors via eigenvalues has been widely studied. This paper focuses on fractional $ID$-$[a, b]$-factor-critical graphs, which are a natural generalization of fractional $[a,b]$-factors. Let $r \ge 1$ be an integer. A graph $G$ is fractional $ID$-$[a, b]$-factor-critical if for every independent set $I$ of $G$ with $|I| = r$, $G - I$ has a fractional $[a, b]$-factor. In 2026, Jia, Fan and Liu posed the spectral version conjecture for a graph to be fractional $ID$-$[a, b]$-factor-critical [Linear Algebra Appl. 732 (2026) 1-17]. In this paper, we first prove the conjecture holds for connected graphs when $b\ge 2r+2$. Furthermore, for minimum degree $δ(G)\ge a+r$, we present spectral radius and size conditions that ensure a graph is fractional $ID$-$[a, b]$-factor-critical, which improve the results of Jia, Fan and Liu.