The sun graph is determined by its signless Laplacian spectrum

Abstract

For a simple undirected graph G, the corresponding signless Laplacian matrix is defined as D(G) + A(G) in which D(G) and A(G) are degree matrix and adjacency matrix of G, respectively. The graph G is said to be determined by its signless Laplacian spectrum, if any graph having the same signless Laplacian spectrum as G is isomorphic to G. Also the Sun graph of order 2n is a cycle Cn with an edge terminating in a pendent vertex attached to each vertex. Among other things, one result in this paper is that the Sun graphs are determined by their signless Laplacian spectrum. Recall that the Laplacian matrix and the signless Laplacian matrix of G are defined as L(G) = D(G) i A(G) and Q(G) = D(G) + A(G), respectively, where D(G) is the diagonal matrix whose diagonal entries are the vertex degrees of G. As it is well-known, the matrices L(G) and Q(G) are positive semi-definite and they have the same characteristic polynomial if and only if G is a bipartite graph. The eigenvalues of the matrices L(G) and Q(G), are denoted by µ1(G) ¸ µ2(G) ¸ . . . ¸ µn(G) = 0 and o1(G) ¸ o2(G) ¸ . . . ¸ on(G), respectively. Furthermore, the second smallest Laplacian eigenvalue of G, µni 1(G), is called the algebraic connectivity, and