On the (Laplacian) spectral radius of bipartite graphs with given number of blocks. (Q2881266)

The authors determine bipartite graphs with a given number \(n\) of vertices and a given number \(k\) of blocks, having either the maximum value of the spectral radius of the adjacency matrix or the spectral radius of the Laplacian matrix. In particular, all such extremal graphs are of the form \(G^{k-1}_{a_1,a_2}\), obtained from the complete bipartite graph \(K_{a_1,a_2}\) by adding \(k-1\) pendant edges to a vertex in the first partition set. In the case of the adjacency matrix, there are either one or two extremal graphs, while in the case of the Laplacian matrix, the unique extremal graph is \(G^{k-1}_{2,n-k-1}\).