On the Transmission-Based Graph Topological Indices

R. Sharafdini, T. Réti

The distance $d(u,v)$ between vertices $u$ and $v$ of a connected graph $G$ is equal to the number of edges in a minimal path connecting them. The transmission of a vertex $v$ is defined by $\sigma(v)=\sum_{u\in V(G)}{d(v,u)}$. A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions $\sigma(u)$ of vertices of $G$. Because $\sigma(u)$ can be derived from the distance matrix of $G$, it follows that transmission-based topological indices form a subset of distance-based topological indices. So far, relatively limited attention has been paid to TT indices, and very little systematic studies have been done. In this paper our aim was i) to define various types of transmission-based topological indices ii) establish lower and upper bounds for them, and iii) determine a family of graphs for which these bounds are best possible. Additionally, it has been shown in examples that using a group theoretical approach the transmission-based topological indices can be easily computed for a particular set of regular, vertex-transitive, and edge-transitive graphs. Finally, it is demonstrated that there exist TT indices which can be successfully applied to predict various physicochemical properties of different organic compounds. Some of them give better results and have a better discriminatory power than the most popular degree-based and distance-based indices (Randić, Wiener, Balaban indices).