Signless Laplacian spectral analysis of a class of graph joins

Abstract

A graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if no other non-isomorphic graph shares the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Every graph of the form K-1 V (C-s U qK(2)), where q >= 1, s >= 3, and the number of vertices is at least 16, is DQS; (2) Every graph of the form K-1 V (C-s1 U C-s2 U & centerdot; & centerdot; & centerdot; U C-st U qK(2)), where t >= 2, q >= 1, si >= 3, and the number of vertices is at least 52, is DQS. Here, K-n and C-n denote the complete graph and the cycle of order n, respectively, while U and V represent the disjoint union and the join of graphs. Moreover, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.