Signless Laplacian characterization of cones over disjoint unions of cycles, edges and isolated vertices

Abstract

Two graphs are said to be Q-cospectral if they share the same signless Laplacian spectrum. A simple graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if there exists no other non-isomorphic simple graph with the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Let G congruent to K1V (CkU qK(2)U sK1), with q, s >= 1, k >= 4, and at least 21 vertices. If k is odd, then G is DQS. Moreover, if k is even and F is Q-cospectral with G, then F congruent to G or F =K1V (C4UPk_3UP3U(q_ 2)K(2)UsK(1)). (2) Let G = K-1 V (Ck(1)U Ck(2)U center dot center dot center dot U Ckt U qK(2)U sK(1)) with t >= 2, q, s >= 1, ki >= 4 and at least 33 vertices. If each k(i) is odd, then G is DQS. (3) The graph K1V ( C3U Ck1 U Ck2 U center dot center dot center dot U Ckt_1 U qK2 U sK1), with t, q, s >= 1 and ki >= 3, is not DQS. Moreover, it is Q-cospectral with K 1 V ( K 1,3 U Ck1 U Ck(2)U center dot center dot center dot U Ckt_1 U qK(2)U ( s _ 1)K-1) . Here P-n, C-n, K(n )and K-n_r,K-r denote the path, the cycle, the complete graph and the complete bipartite graph on n vertices, while U and V represent the disjoint union and the join of two graphs, respectively. Furthermore, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.