The bipartite Turan number and spectral extremum for linear forests

TL;DR

This paper determines the maximum number of edges and spectral radius in bipartite graphs avoiding a linear forest, providing exact extremal graphs for large bipartite graphs.

Contribution

It explicitly calculates the bipartite Turán number for linear forests and characterizes the extremal graphs, also extending results to spectral extremum for such graphs.

Findings

Exact bipartite Turán numbers for linear forests.

Characterization of extremal bipartite graphs avoiding linear forests.

Maximum spectral radius for bipartite graphs without linear forests.

Abstract

The bipartite Tur\'{a}n number of a graph $H$, denoted by $ex(m,n; H)$, is the maximum number of edges in any bipartite graph $G=(X,Y; E)$ with $|X|=m$ and $|Y|=n$ which does not contain $H$ as a subgraph. In this paper, we determined $ex(m,n; F_{\ell})$ for arbitrary $\ell$ and appropriately large $n$ with comparing to $m$ and $\ell$, where $F_\ell$ is a linear forest which consists of $\ell$ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which does not contain $F_{\ell}$ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.