Fractional matching, factors and spectral radius in graphs involving minimum degree

Abstract

A fractional matching of a graph $G$ is a function $f:E(G)\to [0,1]$ such that for any $v\in V(G)$, ${\sum }_{e\in E_G(v)}f(e)\le 1$, where $E_{G}(v)=\{e\in E(G): e~ \mbox{is incident with} ~v~\mbox{in}~G\}$.The fractional matching number of $G$ is ${\mu }_f(G)=\max \{{\sum }_{e\in E(G)}f(e):f$ is a fractional matching of $G\}$. Let $k\in (0,n)$ is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee ${\mu }_f(G)>\frac{n-k}{2}$ in a graph with minimum degree $\delta ,$ which implies the result on the fractional perfect matching due to Fan et al. [Discrete Math. 345 (2022) 112892]. For a set $\{A,B,C,\dots \}$ of graphs, an $\{A,B,C,\dots \}$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ each component of which is isomorphic to one of $\{A,B,C,\dots \}$.We present a tight sufficient condition in terms of the spectral radius for the existence of a $\{K_2,\{C_k\}\}$-factor in a graph with minimum degree $\delta ,$ where $k\ge 3$ is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a $\{K_{1,1},K_{1,2},\dots ,K_{1,k}\}$-factor with $k\ge 2$ in a graph with minimum degree $\delta ,$ which generalizes the result of Miao et al. [Discrete Appl. Math. 326 (2023) 17-32].

12 pages, 3 figures