Fractional matching number and spectral radius of nonnegative matrix of graphs

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 $E_G(v)=\{e\in E(G):e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is ${\mu }_f(G)=\max \{{\sum }_{e\in E(G)}f(e):f$ is fractional matching of $G\}$. For any real numbers $a\ge 0$ and $k\in (0,n)$, it is observed that if $n=|V(G)|$ and $\delta (G)>\frac{n-k}{2}$, then ${\mu }_f(G)>\frac{n-k}{2}$. We determine a function $\phi (a,n,\delta ,k)$ and show that for a connected graph $G$ with $n=|V(G)|$, $\delta (G)\le \frac{n-k}{2}$, spectral radius ${\lambda }_1(G)$ and complement $\overline{G}$, each of the following holds. (i) If ${\lambda }_1(aD(G)+A(G))<\phi (a,n,\delta ,k),$ then ${\mu }_f(G)>\frac{n-k}{2}.$ (ii) If ${\lambda }_1(aD(\overline{G})+A(\overline{G}))<(a+1)(\delta +k-1),$ then ${\mu }_f(G)>\frac{n-k}{2}.$ As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving $Q$-index and $A_{\alpha }$-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.