Two classes of graphs determined by the signless Laplacian spectrum

Abstract

Let Kq, Cq and Pq 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 is DQS) 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 K1∨(Pq∪(n−q−1)K1) is DQS; (2) If n≥21 and 3≤q≤n−1, then K1∨(Cq∪(n−q−1)K1) is DQS if and only if q≠3, where ∪ and ∨ stand for the disjoint union and the join of two graphs, respectively. Moreover, for q=3 in (2) we identify K1∨(K1,3∪(n−5)K1) 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 K1∨Cn−1 and K1∨Pn−1 are DQS.