Signless Laplacian spectral characterization of some disjoint union of graphs

The adjacency matrix of a simple and undirected graph G is denoted by A(G) and D_G is the degree diagonal matrix of G. The Laplacian matrix of G is L(G) = D_G- A(G) and the signless Laplacian matrix of G is Q(G) = D_G+ A(G). The star graph of order n is denoted by S_n. The double starlike treeG_p,n,q is obtained by attaching p pendant vertices to one pendant vertex of the path P_n and q pendant vertices to the other pendant vertex of P_n. In this paper, we first investigate the disjoint union of double starlike graphs G_p,2,q and the star graphs 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 DQS connected non-bipartite graph with n≥ 3 vertices, then G∪ rK₁∪ sK₂ is DQS. Here we give a counterexample for the claim and also we study the graph G∪ rK₁∪ sK₂ for signless Laplacian charcterization when G has at least ((n- 2) (n- 3) + 10) / 2 edges and s= 1. It is shown that the graph K_n∪ K₂∪ rK₁ is DQS for n≥ 4. We also prove that the complement graph of K_n∪ K₂∪ rK₁ is DQS for r> 1 and n≠ 3.