# 11th meeting of the International Academy of Mathematical Chemistry

18-21 June 2015
Kranjska Gora, Slovenia
Europe/Ljubljana timezone
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# Wiener Index under Symmetry Group

Presented by Prof. Ali ASHRAFI
Type: Oral presentation

## Content

The Wiener index has been generalized in many ways, some by considering distance between edges, and others by counting some special types of vertex distances. Graovac and Pisanski \cite{1}, presented an algebraic modification of the Wiener index of a graph by considering its symmetry group. To explain, we assume that $G$ is a graph with automorphism group $\Gamma = Aut(G)$. Define the {\it distance number of an automorphism $g$}, $\delta(g)$, to be the average of $d(u,g(u))$ overall vertices $u \in V(G)$ and $\delta(G) = \frac{1}{|\Gamma||V(G)|}\sum_{u \in V(G)}\sum_{g \in \Gamma}d(u,g(u))$. The {\it modified Wiener index} of $G$ is defined as: $$\hat{W}(G) = \frac{1}{2}|V(G)|^2\delta(G) = \frac{|V(G)|}{2|\Gamma|}\sum _{u \in V(G)}\sum_{g \in \Gamma} d(u, g(u)).$$ Graovac and Pisanski obtained exact formula for this topological index under Cartesian product of graphs. The aim of this talk is to present our recent results on extremal properties of this topological index. Moreover, we present some exact formulas for computing this generalization of the Wiener index under some graph operation \cite{2}. \vskip 3mm \noindent{\bf Keywords:} Wiener index, symmetry group. \vskip 3mm \noindent{\bf AMS Subject Classification Number:} $05C50$. \begin{thebibliography}{10}\label{bibliography} \bibitem{1} A. Graovac and T. Pisanski, On the Wiener index of a graph, {\it J. Math. Chem.} {\bf 8} (1991) 53--62. \bibitem{2} F. Koorepazan-Moftakhar, A. R. Ashrafi, Distance under Symmetry, \textit{MATCH Commun. Math. Comput. Chem.} \textbf{74} (2) (2015) 000-000. \end{thebibliography}

## Place

Location: Kranjska Gora, Slovenia