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 𝐺∪𝑟𝐾₁ is DQS under certain conditions.

We also show that for every 𝑛 > 2, 𝐹𝑛 is not 𝒟-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real ...