A class of unicyclic graphs determined by their Laplacian spectrum

Abstract

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+ 4. 1. Introduction. Let G be a simple graph on n vertices and A(G) be its adja- cency matrix. Let dG(v) be the degree of vertex v in G, and D(G) be the diagonal matrix with the degrees of the corresponding vertices of G on the diagonal and zero elsewhere. Matrix Q(G) = D(G) − A(G) is called the Laplacian matrix of G. The eigenvalues of A(G) (resp., Q(G)) and the spectrum (which consists of eigenvalues) of A(G) (resp., Q(G)) are also called the adjacency (resp., Laplacian) eigenvalues of G and the adjacency (resp., Laplacian) spectrum of G. Since both matrices A(G) and Q(G) are real symmetric matrices, their eigenvalues are all real numbers. So we can assume that �1(G) � �2(G) � ��� � �n(G) and µ1(G) � µ2(G) � ��� � µn(G) = 0 are the adjacency eigenvalues and the Laplacian eigenvalues of G, respectively. Two graphs are adjacency (resp., Laplacian) cospectral if they have the same adjacency (resp., Laplacian) spectrum. Denote by �(G) = �(G;�) = det(�I − A(G)) and �(G;µ) = det(µI − Q(G)) the characteristic polynomial of adjacency matrix and Laplacian matrix of G, respectively. A graph is said to be determined by the adjacency (resp., Laplacian) spectrum if there is no non-isomorphic graph with the same adjacency (resp., Laplacian) spectrum. In general, the spectrum of a graph does not determine the graph and the question Which graphs are determined by their spectrum? ((3)) remains a difficult problem. For the background and some known results about this problem and related topics, we refer the readers to (4) and references therein. For the unicyclic graphs, Haemers