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.

Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

In this paper, we study the connectivity of the complement of a graph. We first establish a simple formula to compute the connectivity of a graph that comes from the join of graphs. Base on this formula, we give an upper and a lower bound to the connectivity of the complement of a graph. Keyword: connectivity, complement, join, vertex of attachment.

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...

The atom graph of a graph is the graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(n log n, nm,n(n+m)) time, which is no more than the complexity of computing the atoms in the gene...

In this paper we derive HF index of some graph operations containing join, Cartesian Product, Corona Product graphs and compute the Y new related to join graphs.