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. 
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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