Signless Laplacian spectral characterization of some disjoint union of graphs

The adjacency matrix of a simple and undirected graph G is denoted by $${\mathcal {A}}(G)$$ A ( G ) and $${\mathcal {D}}_{G}$$ D G is the degree diagonal matrix of G. The Laplacian matrix of G is $${\mathcal {L}}(G)={\mathcal {D}}_{G}-{\mathcal {A}}(G)$$ L ( G ) = D G - A ( G ) and the signless Laplacian matrix of G is $${\mathcal {Q}}(G)={\mathcal {D}}_{G}+{\mathcal {A}}(G) $$ Q ( G ) = D G + A ( G ) . The star graph of order n is denoted by $$S_{n}$$ S n . The double starlike tree $${\mathcal {G}}_{p,n,q}$$ G p , n , q is obtained by attaching p pendant vertices to one pendant vertex of the path $$P_n$$ P n and q pendant vertices to the other pendant vertex of $$P_n$$ P n . In this paper, we first investigate the disjoint union of double starlike graphs $${\mathcal {G}}_{p,2,q}$$ G p , 2 , q and the star graphs $$S_{n}$$ S n for Laplacian (signless) spectral characterization. Also, the signless Laplacian spectral determination of the disjoint union of odd unicyclic graphs and star graphs is studied. Abdian et al. [AKCE Int. J. Graphs Combin. (2018) https://doi.org/10.1016/j.akcej.2018.06.009] proved that if G is a $$\mathcal {DQS}$$ DQS connected non-bipartite graph with $$n\ge 3$$ n ≥ 3 vertices, then $$G\cup rK_{1}\cup sK_{2}$$ G ∪ r K 1 ∪ s K 2 is $$\mathcal {DQS}$$ DQS . Here we give a counterexample for the claim and also we study the graph $$G\cup rK_{1}\cup sK_{2}$$ G ∪ r K 1 ∪ s K 2 for signless Laplacian charcterization when G has at least $$((n-2)(n-3)+10)/2$$ ( ( n - 2 ) ( n - 3 ) + 10 ) / 2 edges and $$s=1$$ s = 1 . It is shown that the graph $$K_{n}\cup K_{2}\cup rK_{1}$$ K n ∪ K 2 ∪ r K 1 is $$\mathcal {DQS}$$ DQS for $$n\ge 4$$ n ≥ 4 . We also prove that the complement graph of $$K_{n}\cup K_{2}\cup rK_{1}$$ K n ∪ K 2 ∪ r K 1 is $$\mathcal {DQS}$$ DQS for $$r>1$$ r > 1 and $$n\ne 3$$ n ≠ 3 .