Geodesic
On graphs with given diameter, number of vertices, and local diversity of balls
Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 1, pp. 65-74

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The $n$-vertex graphs with diameter $d$ and local $t$-diversity of balls, i.e. graphs having $n$ different balls of radius $i$ for every $i\leq t$, in connection with the characterization problem of the diversity vectors of balls of usual connected graphs are studied. For such graphs there exists a lower bound for the number of vertices, defined by the parameters $d$ and $t$. All graphs of the minimal possible order with diameter $d$ and local $t$-diversity of balls (full diversity of balls) are explicitly described up to isomorphism. Moreover, the diversity vector of balls is calculated for any such graph. Ill. 4, bibl. 8.
Keywords: graph, diameter of the graph, metric ball, radius of the ball, number of balls, diversity vector of balls. 
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     title = {On graphs with given diameter, number of vertices, and local diversity of balls},
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T. I. Fedoryaeva. On graphs with given diameter, number of vertices, and local diversity of balls. Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 1, pp. 65-74. http://geodesic.mathdoc.fr/item/DA_2010_17_1_a3/