Wiener, Hyper Wiener and Detour Index of Pseudoregular Graphs

Abstract

The field of graph theory is rich in its theoretical and application area. Drugs and other chemical compounds are often modeled as polygonal shape, where each vertex represents an atom of the molecule and covalent bonds between atoms are represented by edges between the corresponding vertices. This polygonal shape derived from a chemical compound is often called its modecular graph and can be a path, a tree or in general graph. An indicator defined over this molecular graph, the Wiener index \(W(G)\) is defined as \(W(G)=\sum\limits _{u\ne v}d(u,v) \), where the sum is taken through all unordered pairs of vertices of \(G\).  Another indicator is the Hyper Wiener index
\(WW(G)\) is defined as \(WW(G)=\frac{1}{2} \sum\limits _{u\ne v}\left(d(u,v)+d^{2} (u,v)\right)\), where \(d^{2}(u,v)=d(u,v)^{2}\), and the Detour index \(D(G)\) is defined as \(D(G)=\sum\limits _{u\ne v}D(u,v)\), where \(D(u,v)\) denotes the longest distance from \(u\) to \(v\) in \(G\). In this paper, we computed the Wiener, Hyper Wiener, Detour index for a special graph namely, Pseudo-regular graph.