Signless Laplacian spectral analysis of a class of graph joins

A graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if no other non-isomorphic graph shares the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Every graph of the form K1 ∨ (Cs ∪ qK2), where q ≥ 1, s ≥ 3, and the number of vertices is at least 16, is DQS; (2) Every graph of the form K1 ∨(Cs1 ∪Cs2 ∪· · ·∪Cst ∪qK2), where t ≥ 2, q ≥ 1, si ≥ 3, and the number of vertices is at least 52, is DQS. Here, Kn and Cn denote the complete graph and the cycle of order n, respectively, while ∪ and ∨ represent the disjoint union and the join of graphs. Moreover, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.