A new extension of the generalized topological indices (GTI) approach is carried out to represent “simple” and “composite” topological indices (TIs) in an unified way. This approach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randić connectivity indices are expressed by means of ...

The Hosoya polynomial of a graph, H(G, z), has the property that its first derivative, evaluated at z = 1, equals the Wiener index, i.e., W(G) = H’(G, 1). In this paper, an equation is presented that gives the hyper-Wiener index, WW(G), in terms of the first and second derivatives of H(G,z). Also defined here is a hyper-Hosoya polynomial, HH(G,r), which has the property WW(G) = HH’(G, l), analo...

The sum of distances between all the pairs of vertices in a connected graph is known as the {it Wiener index} of the graph. In this paper, we obtain the Wiener index of edge complements of stars, complete subgraphs and cycles in $K_n$.

We study distance-based graph invariants, such as the Wiener index, the Szeged index, and variants of these two. Relations between the various indices for trees are provided as well as formulas for line graphs and product graphs. This allows us, for instance, to establish formulas for the edge Wiener index of Hamming graphs, C4nanotubes and C4-nanotori. We also determine minimum and maximum of ...

By the Von Neumann regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x,y if and only if x+y is a von Neumann regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann regular elements by V nr...

The problem of distances in a graph, G, is one of the most studied questions, both from theoretical point of view and applications (the reader can consult two recent reviews ). It is connected to the Wiener number, W, or "the path number" as denominated by its initiator. In acyclic structures, Wiener number and its extension, hyper-Wiener number, can be defined as edge/path contributions, Ie/p ...

If G is a connected graph with vertex set V , then the eccentric connectivity index of G, ξ(G) is defined as ∑ deg(v).ec(v) where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. The Wiener index W (G) = 1 2 [ ∑ d(u, v)], the hyper-Wiener index WW (G) = 1 2 [ ∑ d(u, v) + ∑ d(u, v)] and the reverseWiener index ∧(G) = n(n−1)D 2 −W (G), where d(u, v) is the distance of two vertice...

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.

Let G be a graph with n vertices and m edges. In many cases the complement of G has the following properties: it is connected, its diameter is 2, its Wiener index is equal to (n 2 ) +m, and its hyper-Wiener index is equal to (n 2 ) + 2m. We characterize the graphs whose complements have the mentioned properties.