Geodesic
Tree-like structure graphs with full diversity of balls
Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 1, pp. 25-41

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Under study is the diversity of metric balls in connected finite ordinary graphs considered as a metric space with the usual shortest-path metric. We investigate the structure of graphs in which all balls of fixed radius $i$ are distinct for each $i$ less than the diameter of the graph. Let us refer to such graphs as graphs with full diversity of balls. For these graphs, we establish some properties connected with the existence of bottlenecks and find out the configuration of blocks in the graph. Using the obtained properties, we describe the tree-like structure graphs with full diversity of balls. Illustr. 8, bibliogr. 22.
Keywords: graph, tree-like structure graphs, metric ball, radius of a ball, number of balls, diversity vector of balls, full diversity of balls. 
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A. A. Evdokimov; T. I. Fedoryaeva. Tree-like structure graphs with full diversity of balls. Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 1, pp. 25-41. http://geodesic.mathdoc.fr/item/DA_2018_25_1_a1/