A note on the spectral characterization of dumbbell graphs

The dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertex-disjoint cycles Ca and Cb joined by a path Pc+3 (c⩾-1) having only its end-vertices in common with the two cycles. By using a new cospectral invariant for (r,r+1)-almost regular graphs, we will show that almost all dumbbell graphs (without cycle C4 as a subgraph) are determined by the adjacency spectrum.