A communication network can be considered to be highly vulnerable to disruption if the failure of few members (nodes or links) can result in no members being able to communicate with very many others‎. ‎These communication networks can be modeled through Graphs‎. ‎There are several Graph-theoretic parameters to describe the stability of Graphs‎. ‎But‎, ‎these parameters are not sufficient to study stability of Graphs‎. ‎This leads to the concept of integrity of a Graph‎. ‎In this paper‎, ‎we obtain the integrity of some Graph operations and some special Graphs which can help us to reconstruct the given network in such a way that it is more stable than the earlier one‎.

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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 Gr, Hr and Gr  Hr, when  is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation (G  H)r = Gr  Hr. We also compute the Wiener index and Wiener polarity index of the skew product.

In this paper, the weighted Szeged indices of Cartesian product and Corona product of two connected Graphs are obtained. Using the results obtained here, the weighted Szeged indices of the hypercube of dimension n, Hamming Graph, C4 nanotubes, nanotorus, grid, t-fold bristled, sunlet, fan, wheel, bottleneck Graphs and some classes of bridge Graphs are computed.

The  rst leap Zagreb index LM1 of a (molecular) Graph, is the sum of squares of the second degrees of vertices (number of their second neighbors), and the second leap Zagreb index LM2 is the sum of the products of the second degrees of pairs of adjacent vertices, and the third leap Zagreb index LM3 is the sum of the product of the degree and second degree of the vertices. In this paper, we determine the  rst, second and third leap Zagreb indices of some Graph operations.

Let G be a simple connected Graph. The first and second Zagreb indices have been introduced as M1(G)= åvÎV(G)degG(v)2 and M2(G)= åuvÎE(G)degG(u)degG(v), respectively, where degG v(degG u) is the degree of vertex v (u). In this paper, we define a new distance-based named Hyper-Zagreb as HM(G) = åe=uvÎE(G)(degG(u) + degG(v))2. In this paper, the Hyper-Zagreb index of the Cartesian product, composition, join and disjunction of Graphs are computed.

The reciprocal degree distance (RDD), defined for a connected Graph G as vertex-degree- weighted sum of the reciprocal distances, that is, RDD (G) = S u, vÎV (G) dG (u)+dG (v)/dG (u, v). The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. In this paper, we present exact formulae for the reciprocal degree distance of join, tensor product, strong product and wreath product of Graphs in terms of other Graph invariants including the degree distance, Harary index, the first Zagreb index and first Zagreb coindex. Finally, we apply some of our results to compute the reciprocal degree distance of fan Graph, wheel Graph, open fence and closed fence Graphs.

Let G(V, E) be a Graph. The common neighborhood Graph (conGraph) of G is a Graph with vertex set V, in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of conGraphs under Graph operations; Graph union, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley Graphs and its conGraphs.

The distance matrix, distance eigenvalue, and distance energy of a connected Graph have been studied in detail in literature where as the study on distance seidel matrix associated with a connected Graph is in progress. The eigenvalues ∂ S 1 ≥ ∂ S 2 >. . . ∂S n of the distance seidel matrix D S (G) of a Graph G forms the distance seidel spectrum of G. We describe here the distance seidel spectrum of some types of subdivision related Graphs of a regular Graph in terms of its adjacency spectrum. We also derive analytic expressions for the distance seidel energy of S¯(Cp), the partial complement of the subdivision Graph of a cycle Cp and the distance seidel energy of S(Cp), the complement of the even cycle C2p