Summary: Let \(G=(P,Q)\) be a bipartite graph and \(G'\) be a graph obtained by joining each vertex of \(P\) and \(Q\) with \(m\) and \(s\) new vertices respectively. We obtain the characteristic, Laplacian and signless Laplacian polynomial of \(G'\). As an application, we give a simple proof for Csikvari’s lemma on eigenvalues of graphs.