Spectral extrema of graphs: Forbidden star-path forests

A path of order $n$ is denoted by $P_n$, and a star of order $n$ is denoted by $S_{n-1}$. A star-path forest is a forest whose connected components are paths and stars. In this paper we determine the maximum spectral radius of graphs that contain no copy of $k{S}_{\mathrm{\ell }-1}\cup P_{\mathrm{\ell }}$, $k_1{S}_{2\mathrm{\ell }-1}\cup k_2{P}_{2\mathrm{\ell }}$ or $k{S}_4\cup 2P_5$ for $n$ appropriately large.