设 H(K_{1,5},P_n,C_l)是由路 P_n的两个悬挂点分别粘上星图K_{1,5}的悬挂点和圈 C_l的点所得的单圈图. 若两个二部图是关于Laplacian 矩阵同谱的, 则它们的线图是邻接同谱的, 两个邻接同谱图含有相同数目的同长闭回路. 如果任何一个与图G关于Laplacian 同谱图都与图G 同构, 那么称图G可由其Laplacian 谱确定. 利用图与线图之间的关系证明了H(K_{1,5},P_n,C_4)、H(K_{1,5},P_n,C_6) 由它们的Laplacian谱确定.

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.