Fractional Matchings in Graphs from the Spectral Radius

Denote by \(\mathcal {G}_{n, \nu ^*}\) ${\mathcal{G}}_{n,{\nu }^{\ast }}$ \((\mathcal {G}^*_{n,\nu ^*})\) $({\mathcal{G}}_{n,{\nu }^{\ast }}^{\ast })$ the collection of all (connected) graphs of order n having a fractional matching number \(\nu ^*\) ${\nu }^{\ast }$ . This paper characterizes the graphs in \(\mathcal {G}_{n,\nu ^*}\) ${\mathcal{G}}_{n,{\nu }^{\ast }}$ and \(\mathcal {G}^*_{n,\nu ^*}\) ${\mathcal{G}}_{n,{\nu }^{\ast }}^{\ast }$ with the maximum spectral radius, and establishes a lower bound for the spectral radius of graphs of order n to guarantee that their fractional matching numbers are at least \(\tau +\frac{1}{2}\) $\tau +\frac{1}{2}$ . In addition, we explore the relationship between the spectral radius, perfect matching and fractional perfect matching of G. Moreover, we present a spectral condition guaranteeing that the matching number of a graph is at least \(k+1\) $k+1$ , which generalizes some previous known results.