Geodesic
A New Proof of the Szeged-Wiener Theorem
Kragujevac Journal of Mathematics, Tome 35 (2011) no. 1, p. 165

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

The Wiener index $W(G)$ is the sum of distances between all pairs of vertices of a connected graph $G$. For an edge $e$ of $G$, connecting the vertices $u$ and $v$, the set of vertices lying closer to $u$ than to $v$ is denoted by $N_e(u)$. The Szeged index, $Sz(G)$, is the sum of products $|N_u(e)| \times |N_v(e)|$ over all edges of $G$. A block graph is a graph whose every block is a clique. The Szeged-Wiener theorem states that $Sz(G) = W(G)$ holds if and only if $G$ is a block graph. A new proof of this theorem if offered, by means of which some properties of block graphs could be established.
Classification : 05C12 05C05
Keywords: Szeged index, Wiener index, Block graph 
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H. Khodashenas; M. J. Nadjafi-Arani; A. R. Ashrafi; I. Gutman. A New Proof of the Szeged-Wiener Theorem. Kragujevac Journal of Mathematics, Tome 35 (2011) no. 1, p. 165 . http://geodesic.mathdoc.fr/item/KJM_2011_35_1_a13/