Graphs determined by signless Laplacian spectra

In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that GrK1sK2 is DQS under certain conditions, where r, s are natural numbers and K1 andK2 denote the complete graphs on one vertex and two vertices, respectively. Applying these results, some DQS graphs with independent edges and isolated vertices are obtained.