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The hyper-Wiener index is an important topological invariant, which is based on the distance between two vertexes on a graph. This paper is concerned with the hyper-Wiener indices of Vicsek networks. Based on the method of finite pattern, we calculate the asymptotic formula of the hyper-Wiener indices of Vicsek networks.

The hyper-Wiener index on a graph is an important topological invariant that is defined as one half of the sum of the distances and square distances between all pairs of vertices of a graph. In this paper, we develop the discrete version of finite pattern to compute the accurate formulas of the hyper-Wiener indices of the Sierpiński skeleton networks.

The hyper-Wiener index plays an important role in chemical graph theory. In this paper, using the technique named finite pattern, we discuss the hyper-Wiener index on level-3 Sierpinski gasket which is a self-similar fractal.

The pent–heptagonal carbon nanosheets (CNSs) admit significant industrial applications. We shall compute the distance-based Hosoya polynomial of CNSs including $V{C}_5{C}_7$ and $H{C}_5{C}_7$. The first-, second- and third-order derivatives of Hosoya polynomial, respectively, lead to the Wiener, hyper-Wiener and Tratch–Stankevitch–Zefirov (TSZ) indices which are the distance-based topological indices. These indices have many physical applications and can be correlated to physio-chemical properties of chemical structures. Results exhibit that despite the fact that both sheets have the same chemical formula, they have different Hosoya polynomials and related distance-based topological indices, resulting in distinct quantitative structure–property relation (QSPR) and quantitative structure–activity relation (QSAR) due to configuration differences.

Let $R$ be a ring with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is an undirected graph with all the elements of $R$ as vertices and where distinct vertices $x$, $y$ are adjacent if and only if $x+y$ is a unit of $R$. In this paper, we investigate the Wiener index and hyper-Wiener index of $G(R)$ and explicitly determine their values.

The Wiener number $W(G)$ of a graph $G$ was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by $WW(G)$ and defined as $WW(G)=\frac{1}{2}{\sum }_{u\in V}[d(u|G)+d^2(u|G)]$. In this paper, we establish some upper and lower bounds for $WW(G)$, in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

The kth power of a graph G, denoted by G^k, is a graph whose vertex set is V(G), two distinct vertices being adjacent in G^k if and only if their distance in G is at most k. The hyper-Wiener index WW(G) of a graph G is defined as , where d_G(u,v) is the distance between vertices u and v in G. In this paper, the bounds on the hyper-Wiener index of the graph G^k are given. The Nordhaus–Gaddum-type inequality for the hyper-Wiener-index of the graph G^k is also presented.

Topological indices, i.e., numerical invariants suitably associated to graphs and only depending upon their isomorphism type, have important applications in Chemistry. Their computation constitutes an important branch of Chemical Graph Theory. In this paper, we focus on some degree and distance-based invariants related to the Zagreb indices, the Szeged index and the Wiener index, namely, the F-index, the vertex PI index and the hyper-Wiener index. In particular, we find the formulæ to compute these topological invariants for wreath product of graphs.