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Given a graph G . The fewest number of vertices whose deletion yields a bipartite graph from G was defined by S. Fallat and Yi-Zheng Fan to be the vertex bipartiteness of G and it is denoted by υ b ( G ) . We consider the set Σ k ( n ) defined by { G = ( V ( G ) , E ( G ) ) : G connected , | V ( G ) | = n and υ b ( G ) ≤ k } . In this work we identify the graph in Σ k ( n ) with maximum spectral radius and maximum signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

Let k , p , q be positive integers with k < p < q + 1 . We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph K p , q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q . Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences. [ABSTRACT FROM AUTHOR]