In this paper, we introduced new index from the Zagreb family, named as -index is defined sum of degree five all vertices a graph. We derive some different graph operations such that Join, Cartesian product, Composition, Corona Tensor Product, Strong Disjunction, Symmetric difference, join Subdivision vertex are obtained.

A graph is said to be total-colored if all the edges and vertices of are colored. path in a total proper (i) any two adjacent on differ color, (ii) internal (iii) vertex differs color from its incident path. called total-proper connected by graph. For [Formula: see text], connection number denoted defined as smallest colors required make text] connected. In this paper, we study for operations. ...

The sigma coindex is defined as the sum of squares differences between degrees all nonadjacent vertex pairs. In this paper, we propose some mathematical properties coindex. Later, present precise results for coindices various graph operations such tensor product, Cartesian lexicographic disjunction, strong union, join, and corona product.

In this paper, the Hyper - Zagreb index of the Cartesian product, composition and corona product of graphs are computed. These corrects some errors in G. H. Shirdel et al.[11].

let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.