the schultz index for product graphs

Among the binary operations made with graphs‎, ‎the cartesian‎, ‎corona‎, ‎and lexicographic are three wellknown products‎, ‎as well as the cartesian sum‎. ‎topological indices are graph invariants used to describe graphs associated with molecules‎, ‎one of these is the schultz index‎, ‎which can be obtained as‎ ‎ ‎∑ u≠v (deg u+ deg v) d(u,v), ‎where the sum runs over all pairs of distinct vertices of the graph‎. ‎in this paper‎, ‎we give explicit expressions for the schultz index of cartesian and corona‎, ‎with alternative proofs to those given in the literature‎, ‎as well as for lexicographic product and the cartesian sum‎, ‎all of these formulas involve order and size of factors‎, ‎additionally‎, ‎the first three involve both wiener and schultz indices of factors‎, ‎corona and lexicographic also involve zagreb index and the last one just zagreb.‎ ‎