Extremal graphs with bounded vertex bipartiteness number

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 υb(G). We consider the set Σk(n) defined by{G=(V(G),E(G)):G connected,|V(G)|=n and υb(G)≤k}. In this work we identify the graph in Σk(n) with maximum spectral radius and maximum signless Laplacian spectral radius.