The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

It is shown that the Shields–Harary index of vulnerability of the complete bipartite graph Km,n, with respect to the cost function f (x)= 1− x, 0 x 1, is m, if n m+ 2√m, and 1 n+1 (n+m) 2 4 , ifm n<m+ 2 √ m. It follows that the Shields–Harary number ofKm,n with respect to any concave continuous cost function f on [0, 1] satisfying f (1)=0 ismf (0), if n m+2√m, and between 1 n+1 (n+m) 2 4 f (0) ...

A {it topological index} of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. 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.

We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.