In studying branching properties of alkanes, several numbering schemes for the edges of the associated hydrogen-suppressed graph were proposed based on the degrees of the end vertices of an edge [11]. To preserve rankings of certain molecules, some inequalities involving the weights of edges needed to be satisfied. Randić [11] stated that weighting all edges uv of the associated graph G by (d(u...

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of the product degrees adjacent vertices G. In this paper, we introduce several transformations that are useful tools for study extremal properties index. Using these and symmetric structural representations some graphs, determine graphs having maximal with prescri...