‎the harmonic index of a connected graph $g$‎, ‎denoted by $h(g)$‎, ‎is‎ ‎defined as $h(g)=sum_{uvin e(g)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in g‎. ‎in this paper‎, ‎expressions for the harary indices of the‎ ‎join‎, ‎corona product‎, ‎cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

In this paper, we present the various upper and lower bounds for the product version of reciprocal degree distance in terms of other graph inavriants. Finally, we obtain the upper bounds for the product version of reciprocal degree distance of the composition, Cartesian product and double of a graph in terms of other graph invariants including the Harary index and Zagreb indices. .

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paperweprovide anupper boundof theHarary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with themaximumHarary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In ad...

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, λ a real number, and Wλ(G) = ∑ k≥1 d(G, k)kλ. Wλ(G) is called the Wiener-type invariant of G associated to real number λ. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyperWiener index and Tratc...

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based indices such as Wiener index, hyper-Wiener Harary Tratch – Stankevitch Zefirov index. In present study, we determine closed form for third type chain hex-derived network dime...

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

a topological index of a molecular graph g is a numeric quantity related to g which isinvariant under symmetry properties of g. in this paper we obtain the randić, geometricarithmetic,first and second zagreb indices , first and second zagreb coindices of tensorproduct of two graphs and then the harary, schultz and modified schultz indices of tensorproduct of a graph g with complete graph of ord...

We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e. a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating seri...

The stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. As the network begins losing links or nodes, eventually there is a loss in its effectiveness. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...

in accordance with the theorem in Katz [1955]. This result was anticipated in turn by Rubin and Sitgreaves in an unpublished memorandum cited in Katz [1955]. In order to contribute something positive in this note, we now prove the theorem about graphs and groups which justifies the formula given in Harary and Read [1966] for the number of ways to label a structure. Since this is a sequel to Har...