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...

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.

Let $G$ be a molecular graph with vertex set $V(G)$, $d_G(u, v)$ the topological distance between vertices $u$ and $v$ in $G$. The Hosoya polynomial $H(G, x)$ of $G$ is a polynomial $sumlimits_{{u, v}subseteq V(G)}x^{d_G(u, v)}$ in variable $x$. In this paper, we obtain an explicit analytical expression for the expected value of the Hosoya polynomial of a random benzenoid chain with $n$ hexagon...

let $g$ be a molecular graph with vertex set $v(g)$, $d_g(u, v)$ the topological distance between vertices $u$ and $v$ in $g$. the hosoya polynomial $h(g, x)$ of $g$ is a polynomial $sumlimits_{{u, v}subseteq v(g)}x^{d_g(u, v)}$ in variable $x$. in this paper, we obtain an explicit analytical expression for the expected value of the hosoya polynomial of a random benzenoid chain with $n$ hexagon...