On the Terminal Wiener and Edge-Wiener Indices of an Infinite Family of Dendrimers

Ali Iranmanesh
Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran

Mahdieh Azari
Islamic Azad University, Kazerun, Iran



Content: Let $G$ be a simple connected $n$-vertex graph with the vertex set $V(G)=\{ v_{1} ,v_{2} ,...,v_{n} \}$ and let ${\rm P} =(p_{1} ,p_{2} ,...,p_{n} )$ be an $n$-tuple of nonnegative integers. The thorn graph $G_{{\rm P} }$ is the graph obtained by attaching $p_{i}$ pendent vertices to the vertex $v_{i} $ of $G$, for $i=1,2,...,n$. The concept of thorn graphs was first introduced by Gutman [1] and eventually found a variety of chemical applications. The terminal Wiener index $TW(G)$ of $G$ [2] is defined as the sum of distances between all pairs of pendent vertices of $G$ and the edge-Wiener index $W_e(G)$ of $G$ [2-4] is defined as the sum of distances between all pairs of its edges. Two possible distances $d_0(e,f|G)$ and $d_4(e,f|G)$ between the edges $e$ and $f$ of $G$ can be considered and according to them, the first edge-Wiener index $W_{e_0}(G)$ and the second one $W_{e_4}(G)$ are introduced. In this paper, we present exact formulae for computing the terminal Wiener index, the first and the second edge-Wiener indices of an infinite family of dendrimers by considering them as the thorn graphs of simpler dendrimers. %------------------------------------------- \vspace{5mm} \noindent{\bf References:} \begin{itemize} \item[{[1]}] I. Gutman, Distance in thorny graph, \textit{Publ. Inst. Math. (Beograd)}, \textbf{63} (1998) 31-36. %-------------------------------------------------------------------- \item[{[2]}] I. Gutman, B. Furtula, M. Petrovi\'c, Terminal Wiener index, \textit{J. Math. Chem.} \textbf{46} (2009) 522-531. %-------------------------------------------------------------------- \item[{[3]}]P. Dankelman, I. Gutman, S. Mukwembi, H.C. Swart: The edge Wiener index of a graph, \textit{Discrete Math.} \textbf{309}(10) (2009) 3452-3457 %-------------------------------------------------------------------- \item[{[4]}] A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani, The edge versions of the Wiener index, \textit{MATCH Commun. Math. Comput. Chem.} \textbf{61} (2009) 663-672. %-------------------------------------------------------------------- \item[{[5]}] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, \textit{European J. Combin.} \textbf{30} (2009) 1149-1163. \end{itemize}

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