Spectral conditions for forbidden subgraphs in bipartite graphs

TL;DR

This paper investigates spectral radius conditions for bipartite graphs to exclude certain subgraphs, providing new proofs and extending results to include all $T_{2k+3}$ trees, and identifying extremal outerplanar bipartite graphs.

Contribution

It offers a simpler proof for a known spectral condition for cycles, extends the spectral condition to all $T_{2k+3}$ trees, and determines the maximum spectral radius among large outerplanar bipartite graphs.

Findings

Proved that bipartite graphs with spectral radius at least that of $K_{k,n-k}$ contain all $T_{2k+3}$ trees.

Provided a new, simpler proof for the spectral condition involving $C_{2k+2}$ cycles.

Identified $K_{1,n-1}$ as the extremal outerplanar bipartite graph with maximum spectral radius for large $n$.

Abstract

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-Tur\'{a}n 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 $\rho(G)\geq \rho(K_{k,n-k})$, 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 $\rho(G)\geq \rho(K_{k,n-k})$, 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.