Signless laplacian spectral characterization of roses

Abstract

A p-rose graph Γ = RG(a3, a4, . . . , as) is a graph consisting of p =a3 + a4 + · · · + as ≥ 2 cycles that all meet in one vertex, and ai (3 ≤ i ≤ s) is the number of cycles in Γ of length i. A graph G is said to be DLS (resp., DQS) if it is determined by the spectrum of its Laplacian (resp. signless Laplacian) matrix, i. e. if every graph with the same spectrum is isomorphic to G. He and van Dam [10] recently proved that all p-roses, except for two non-isomorphic exceptions, are DLS. In this paper we show that for p ≥ 3 all p-roses are DQS.