Spectral conditions for forbidden subgraphs in bipartite graphs

A graph $G$ is $H$-free, if it contains no $H$ as a subgraph. A graph is said to be \emph{$H$-minor free}, if it does not contain $H$ as a minor. In recent years, Nikiforov asked that what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we consider about some Brualdi-Solheid-Turán type problems on bipartite graphs. In 2015, Zhai, Lin and Gong proved that if $G$ is a bipartite graph with order $n \geq 2k+2$ and $ρ(G)\geq ρ(K_{k,n-k})$, then $G$ contains a $C_{2k+2}$ unless $G \cong K_{k,n-k}$ [Linear Algebra Appl. 471 (2015)]. Firstly, we give a new and more simple proof for the above theorem. Secondly, we prove that if $G$ is a bipartite graph with order $n \geq 2k+2$ and $ρ(G)\geq ρ(K_{k,n-k})$, then $G$ contains all $T_{2k+3}$ unless $G \cong K_{k,n-k}$. Finally, we prove that among all outerplanar bipartite graphs on $n>344569$ vertices, $K_{1,n-1}$ attains the maximum spectral radius.