The bipartite Turan number and spectral extremum for linear forests

Abstract

The bipartite Turán 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 ℓ ) for arbitrary ℓ and appropriately large n with comparing to m and ℓ, where F ℓ is a linear forest which consists of ℓ 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 ℓ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.