Distance spectral conditions for $ID$-factor-critical and fractional $[a, b]$-factor of graphs

Abstract

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive integers $a$ and $b$ with $a\leq b$, let $h$: $E(G)\rightarrow [0, 1]$ be a function on $E(G)$ satisfying $a\leq\sum _{e\in E_{G}(v_{i})}h(e)\leq b$ for any vertex $v_{i}\in V(G)$. Then the spanning subgraph with edge set $E_{h}$, denoted by $G[E_{h}]$, is called a fractional $[a, b]$-factor of $G$ with indicator function $h$, where $E_{h}=\{e\in E(G)\mid h(e)>0\}$ and $E_{G}(v_{i})=\{e\in E(G)\mid e$ is incident with $v_{i}$ in $G$\}. A graph is defined as a fractional $[a, b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a, b]$-factor. For any integer $k\geq 1$, a graph has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper, we firstly give a distance spectral radius condition of $G$ to guarantee that $G$ is $ID$-factor-critical. Furthermore, we provide sufficient conditions in terms of distance spectral radius and distance signless Laplacian spectral radius for a graph to contain a fractional $[a, b]$-factor, fractional $[a, b]$-deleted-factor and $k$-factor.