The maximum spectral radius of irregular bipartite graphs

TL;DR

This paper determines the asymptotic maximum spectral radius of irregular bipartite graphs with maximum degree three, characterizes graphs with large maximum degree, and provides bounds for general cases.

Contribution

It establishes the asymptotic value of the maximum spectral radius for subcubic bipartite graphs and characterizes extremal graphs for large maximum degree.

Findings

Asymptotic maximum spectral radius is $3- heta(rac{ ext{pi}^2}{n^2})$ for subcubic bipartite graphs.

Characterization of irregular bipartite graphs with maximum spectral radius when maximum degree is at least $loor{n/2}$.

Upper bounds on spectral radius based on order and maximum degree.

Abstract

A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{\pi^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum degree, we present an upper bound on the spectral radius of irregular bipartite graphs in terms of the order and maximum degree.