Abstract
The dumbbell graph, denoted by D a , b , c , is a bicyclic graph consisting of two vertex-disjoint cycles C a and C b joined by a path P c + 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 C 4 as a subgraph) are determined by the adjacency spectrum.
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