mathematical properties and computations of banahatti indices for a nano-structure toroidal polyhex network

Let g be the connected graph with vertex set v(g) and edge set e(g).the first and second k banhatti indices of g are defined as b1(g)=σue[dg (u) +dg (e)] and b2(g)=σue[dg (u) +dg (e)]  where ue means that the vertex u and edge e are incident in g.the first and second k hyper banhatti indices of g are defined as hb1(g) = σue[dg(u) + dg (e)]2 and hb2(g) = σue[dg(u) dg (e)]2 respectively . in this paper, we compute the first and second k banhatti indices of toroidal polyhex network. in addition, the first and second k hyper banhatti indices of toroidal polyhex networks are determined.