the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...

In this paper, we present sharp bounds for the Zagreb indices, Harary index and hyperWiener index of graphs with a given matching number, and we also completely determine the extremal graphs. © 2010 Elsevier Ltd. All rights reserved.

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.