In general, the relations among Zagreb polynomials on three graph operators are discussed in this paper. Specifically, relations between Zagreb polynomials of a graph G and a graph obtained by applying the operators S(G), R(G) and Q(G) are investigated. In a separate section, the relation between Zagreb polynomial of a graph G and its corona is also described.

Topological indices play a very important role in the mathematical chemistry. The topological indices are numerical parameters of a graph. The degree sequence is obtained by considering the set of vertex degree of a graph. Graph operators are the ones which are used to obtain another broader graphs. This paper attempts to find degree sequence of vertex– F join operation of graphs for some standard graphs.