Spectral Characterizations of Dumbbell Graphs

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertexdisjoint cycles Ca, Cb and a path Pc+3 (c > −1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) > 3, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707–1714]. In particular we show that Da,b,0 with 3 6 gcd(a, b) < a or gcd(a, b) = a and b 6= 3a is determined by the spectrum. For b = 3a, we determine the unique graph cospectral with Da,3a,0. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs.