Two classes of graphs determined by the signless Laplacian spectrum

Let K-q, C-q and P-q denote the complete graph, the cycle and the path with q vertices, respectively. We use Q(G) to denote the signless Laplacian matrix of a simple undirected graph G, and say that G is determined by its signless Laplacian spectrum (for short, G isDQS) if there is no other non-isomorphic graph with the same signless Laplacian spectrum. In this paper, we prove the following results: (1) If n >= 21 and 0 <= q <= n-1, then K1V(P-q boolean OR(n-q-1)K-1) is DQS; (2) If n >= 21 and 3 <= q <= n-1, then K-1 boolean OR(C-q boolean OR(n-q-1)K-1) is DQS if and only if q not equal 3, where boolean OR and boolean OR stand for the disjoint union and the join of two graphs, respectively. Moreover, for q=3 in (2) we identify K-1 boolean OR(K-1,K-3 boolean OR(n-5)K-1) as the unique graph sharing the signless Laplacian spectrum with the graph under consideration. Our results extend results of [Czechoslovak Math. J. 62 (2012) 1117-1134] and [Czechoslovak Math. J. 70 (2020) 21-31], where the authors showed that K-1 boolean OR Cn-1 and K-1 boolean OR Pn-1 are DQS. (c) 2024 Elsevier Inc. All rights are reserve including those for text and data mining, AI training, and similar technologies.