Abstract:

Let H(K_{1,5},P_n,C_l) be a unicyclic graph obtained from a path P_n by attaching a star K_{1,5} and a cyclic C_l to its two pendent vertices respectively. If two bipartite graphs are Laplacian cospectral, then their line graphs are adjacency cospectral. The numbers of the same length walks are equal in two adjacency cospectral graphs. A graph G is called to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.  Using the relation between graphs and line graphs, it is proved that the unicyclic graphs H(K_{1,5},P_n,C_4), H(K_{1,5},P_n,C_6) are determined by their Laplacian spectra.

Key words: Laplacian spactrum, line graph, adjacency spactrum, even cyclic