Geodesic
The Wiener Index for Graphs of Arbitrary Girth and Their Edge Graphs
Sibirskij žurnal industrialʹnoj matematiki, Tome 12 (2009) no. 4, pp. 44-50

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We consider the invariant $W(G)$ (Wiener index) of a simple connected nondirected graph $G$, which is equal to the sum of distances between all pairs of vertices in the natural metric. We show that for every $g\ge5$ there exist planar graphs $G$ with a shortest cycle of length $g$ for which $W(L(G))=W(G)$, where $L(G)$ is the edge graph for $G$.
Mots-clés : invariant graph
Keywords: distance in graphs, Wiener index. 
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     author = {A. A. Dobrynin},
     title = {The {Wiener} {Index} for {Graphs} of {Arbitrary} {Girth} and {Their} {Edge} {Graphs}},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {44--50},
     publisher = {mathdoc},
     volume = {12},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2009_12_4_a4/}
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A. A. Dobrynin. The Wiener Index for Graphs of Arbitrary Girth and Their Edge Graphs. Sibirskij žurnal industrialʹnoj matematiki, Tome 12 (2009) no. 4, pp. 44-50. http://geodesic.mathdoc.fr/item/SJIM_2009_12_4_a4/