On a Laplacian spectral characterization of graphs of index less than 2

A graph is said to be determined by the adjacency (respectively, Laplacian) spectrum if there is no other non-isomorphic graph with the same adjacency (respectively, Laplacian) spectrum. The maximum eigenvalue of A(G) is called the index of G. The connected graphs with index less than 2 are known, and each is determined by its adjacency spectrum. In this paper, we show that graphs of index less than 2 are determined by their Laplacian spectrum.