Fractional matching, factors and spectral radius in graphs involving minimum degree (2304.12049v1)

Abstract: A fractional matching of a graph $G$ is a function $f:E(G)\rightarrow [0, 1]$ such that for any $v\in V(G)$, $\sum_{e\in E_{G}(v)}f(e)\leq1$, where $E_{G}(v)={e\in E(G): e~ \mbox{is incident with} ~v~\mbox{in}~G}$.The fractional matching number of $G$ is $\mu_{f}(G)=\mathrm{max}{\sum_{e\in E(G)}f(e):f$ 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 ${A, B, C, \ldots}$ of graphs, an ${A, B, C, \ldots}$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ each component of which is isomorphic to one of ${A, B, C, \ldots}$.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 $k\geq 3$ is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a ${K_{1, 1}, K_{1, 2}, \ldots , K_{1, k}}$-factor with $k\geq2$ in a graph with minimum degree $\delta,$ which generalizes the result of Miao et al. [Discrete Appl. Math. 326 (2023) 17-32].