The spectral radius of graphs with fractional matching number

Let ${\mathcal{G}}_{n,{\beta }^{\ast }}$ $({\mathcal{G}}_{n,{\beta }^{\ast }}^{\ast })$ be the set of all (connected) graphs of order $n$ with fractional matching number ${\beta }^{\ast }$. In this paper, the graphs with maximal spectral radius in ${\mathcal{G}}_{n,{\beta }^{\ast }}$ and ${\mathcal{G}}_{n,{\beta }^{\ast }}^{\ast }$ are characterized, respectively. Moreover, a lower bound for the spectral radius in graphs with order $n$ to guarantee the existence of a perfect fractional matching is also given, which generalizes the main result of O [Suil O, Spectral radius and matchings in graphs, Linear Algebra and its Applications, 2020].