Two classes of graphs determined by the signless Laplacian spectrum

Abstract

Let $K_q$, $C_q$ and $P_q$ denote the complete graph, the cycle and the path with q vertices, respectively. We use $Q\left(G\right)$ to denote the signless Laplacian matrix of a simple undirected graph G, and say that G is determined by its signless Laplacian spectrum (for short, G is DQS) if there is no other non-isomorphic graph with the same signless Laplacian spectrum. In this paper, we prove the following results:

• (1)
  If $n\ge 21$ and $0\le q\le n-1$, then $K_1\vee (P_q\cup (n-q-1\left)K_1\right)$ is DQS;
• (2)
  If $n\ge 21$ and $3\le q\le n-1$, then $K_1\vee (C_q\cup (n-q-1\left)K_1\right)$ is DQS if and only if $q\ne 3$,

where ∪ and ∨ stand for the disjoint union and the join of two graphs, respectively. Moreover, for $q=3$ in (2) we identify $K_1\vee (K_{1,3}\cup (n-5\left)K_1\right)$ as the unique graph sharing the signless Laplacian spectrum with the graph under consideration. Our results extend results of [Czechoslovak Math. J. 62 (2012) 1117–1134] and [Czechoslovak Math. J. 70 (2020) 21–31], where the authors showed that $K_1\vee C_{n-1}$ and $K_1\vee P_{n-1}$ are DQS.