Spectral characterizations of dumbbell graphs

A dumbell graph, denoted by D-a,D-b,D-c, is a bicyclic graph consisting of two vertex-disjoint cycles C-a, C-b 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 D-a,D-b,D-0 (without cycles C-4) with ged (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 D-a,D-b,D-0 with 3 <= gcd (a,b) < a or gcd (a,b) = a and b not equal 3a is determined by the spectrum. For b = 3a, we determine the unique graph cospectral with D-a,D-3a,D-0. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs.