The bipartite Turan number and spectral extremum for linear forests

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;Fl) for arbitrary l and appropriately large n with comparing to m and l, where Fl 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 Fl as a subgraph and characterize all extremal graphs which attain the maximum spectral radius. AMS Classification: 05C35; 05C50; 05C38

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