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 $a\le b$, let $h$: $E(G)\to [0,1]$ be a function on $E(G)$ satisfying $a\le {\sum }_{e\in E_G(v_i)}h(e)\le b$ 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 $k\ge 1$, 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.

10 pages