Geodesic
On a Graph Invariant Related to the sum of all Distances in a Graph
Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 18

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Let $W(G)$ be the sum of distances between all pairs of vertices of a graph $G$. For an edge $e$ of $G$, connecting the vertices $u$ and $v$, the number $n_u(e)$ counts the vertices of $G$ that lie closer to $u$ than to $v$. In this paper we consider the graph invariant $W^\ast(G)=\sum_e n_u(e)n_v(e)$, defined for any connected graph $G$. According to a long-known result in the theory of graph distances, if $G$ is a tree then $W^\ast(G)=W(G)$. We establish a number of properties of the graph invariant $W^\ast$.
Classification : 05C12 
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     author = {A. Dobrynin and Ivan Gutman},
     title = {On a {Graph} {Invariant} {Related} to the sum of all {Distances} in a {Graph}},
     journal = {Publications de l'Institut Math\'ematique},
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     year = {1994},
     language = {en},
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A. Dobrynin; Ivan Gutman. On a Graph Invariant Related to the sum of all Distances in a Graph. Publications de l'Institut Mathématique, _N_S_56 (1994) no. 70, p. 18 . http://geodesic.mathdoc.fr/item/PIM_1994_N_S_56_70_a2/