On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

Reciprocal version of product degree distance of cactus graphs

The reciprocal version of product degree distance is a product degree weighted version of Harary index defined for a connected graph G as RDD*(G) = Sigma({x, y}subset of V(G) )(d(G)(x).d(G)(y))/d(G)(x,y), where d(G)(x) is the degree of the vertex x and d(G)(x, y) is the distance from x to y in G. This article is attain the value of RDD* of different types of cactus such as triangular, square and hexagonal chain cactus graphs.Publisher's Versio

Degree Distance and Gutman Index of Two Graph Products

The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph

Moments in graphs

Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\em $\rho$-moment} of $G$ at vertex $u$ is defined to be M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot) stands for the distance function. Adding up all these numbers, we obtain the {\em $\rho$-moment of $G$}: M_G^{\rho}=\sum_{u\in V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $\rho(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $\rho(u)=\delta(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $\rho$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $\rho$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $\rho$-moment for every $\rho$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product

Product version of reciprocal degree distance of composite graphs

In this paper‎, ‎we present the upper bounds for the product version of reciprocal degree distance of the tensor product‎, ‎join and strong product of two graphs in terms of other graph invariants including the Harary index and Zagreb indices‎

Moments in graphs

Let G be a connected graph with vertex set V and a weight function that assigns a nonnegative number to each of its vertices. Then, the -moment of G at vertex u is de ned to be M G(u) = P v2V (v) dist(u; v), where dist( ; ) stands for the distance function. Adding up all these numbers, we obtain the -moment of G: This parameter generalizes, or it is closely related to, some well-known graph invari- ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the degree distance D0(G), obtained when (u) = (u), the degree of vertex u. In this paper we derive some exact formulas for computing the -moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding -moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same -moment for every (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.Postprint (author’s final draft