The Cartesian product and join are two classical operations in graphs. Let dL(G)(e) be the degree of a vertex e in line graph L(G) of a graph G. The edge versions of atom-bond connectivity (ABCe) and geometric arithmetic (GAe) indices of G are defined as ∑ef∈E(L(G))dL(G)(e)+dL(G)(f)-2dL(G)(e)×dL(G)(f) and ∑ef∈E(L(G))2dL(G)(e)×dL(G)(f)dL(G)(e)+dL(G)(f), respectively. In this paper, ABCe and GAe indices for certain Cartesian product graphs (such as Pn□Pm, Pn□Cm and Pn□Sm) are obtained. In addition, ABCe and GAe indices of certain join graphs (such as Cm+Pn+Sr, Pm+Pn+Pr, Cm+Cn+Cr and Sm+Sn+Sr) are deduced. Our results enrich and revise some known results.
Keywords: Cartesian product graph; atom-bond connectivity index; geometric arithmetic index; join graph; line graph.