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 ≅ K 1 ∨ C k ∪ q K 2 ∪ s K 1 , $G\cong {K}_{1}\vee \left({C}_{k}\cup q{K}_{2}\cup s{K}_{1}\right),$ 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 ≅ G or F ≅ K 1 ∨ C 4 ∪ P k − 3 ∪ P 3 ∪ ( q − 2 ) K 2 ∪ s K 1 . $$F\cong G\quad \text{or}\quad F\cong {K}_{1}\vee \left({C}_{4}\cup {P}_{k-3}\cup {P}_{3}\cup \left(q-2\right){K}_{2}\cup s{K}_{1}\right).$$ (2) Let G ≅ K 1 ∨