Additionally, research on planar graphs has yielded new insights into the anti-Ramsey numbers for paths and cycles, thereby bridging classical graph theory with geometric constraints [3].

Abstract. 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.