Geodesic
Majorants and minorants in the graph class with given number of vertices and diameter
Diskretnyj analiz i issledovanie operacij, Tome 20 (2013) no. 1, pp. 58-76.

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Majorants (minorants), i.e., extremal graphs such that for any $i\ge0$ exact upper (lower) estimates for the number of different balls of the radius $i$ are attained at, are studied in the class of the $n$-vertex graphs with diameter $d$. For all parameters $n$ and $d$, the minorants are described explicitly. It is found out when the majorants exist in the class of $n$-vertex graphs with diameter $d$, and the corresponding extremal graphs are described. Il. 9, bibliogr. 8.
Keywords: graph, metric ball, radius of the ball, the number of balls, estimate of the number of balls, extremal graph. 
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T. I. Fedoryaeva. Majorants and minorants in the graph class with given number of vertices and diameter. Diskretnyj analiz i issledovanie operacij, Tome 20 (2013) no. 1, pp. 58-76. http://geodesic.mathdoc.fr/item/DA_2013_20_1_a5/

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