If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index We of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on We in terms of order and size. In particular we prove the asymptotically sharp upper bound We(G) ≤ 25 55 n5 + O(n9/2) for gr...

For an edge uv of a graph G, the weight of the edge e = uv is denoted by w(e) = 1/ √ d(u)d(v). Then

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. In the paper, we give a sharp lower bound on the Randić index of cacti.

The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) of all edges uv of G, where α is a real number (α 6= 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T ). Also, for −1 ≤ α < 0, we give a sharp lower bound and a upper bound for wα(T ).

 Mobile sensor networks rely heavily on inter-sensor connectivity for collection of data. Nodes in these networks monitor different regions of an area of interest and collectively present a global overview of some monitored activities or phenomena. A failure of a sensor leads to loss of connectivity and may cause partitioning of the network into disjoint segments. A number of approaches have be...

The Wiener index W (G) of a connected graph G is defined to be the sum

Let G be a graph and δv the degree of its vertex v . The connectivity index of G is χ = ∑ (δu δv) −1/2 , with the summation ranging over all pairs of adjacent vertices of G . We offer a simple proof that (a) among n-vertex graphs without isolated vertices, the star has minimal χvalue, and (b) among n-vertex graphs, the graphs in which all components are regular of non–zero degree have maximal (...

The atom-bond connectivity (ABC) index, introduced by Estrada et al. in 1998, displays an excellent correlation with the formation heat of alkanes. We give upper bounds for this graph invariant using the number of vertices, the number of edges, the Randić connectivity indices, and the first Zagreb index. We determine the unique tree with the maximum ABC index among trees with given numbers of v...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...