Geodesic
Exact upper estimates of the number of different balls of given radius for the graphs with fixed number of vertexes and diameter
Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 6, pp. 74-92.

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For the $n$-vertex usual connected graphs with diameter $d$ the exact upper estimates of the number of different balls of given radius are obtained from investigation of the location of the centers of distinct balls. Ill. 5, bibl. 7.
Keywords: graph, diameter of the graph, metric ball, radius of the ball, number of balls, estimates. 
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T. I. Fedoryaeva. Exact upper estimates of the number of different balls of given radius for the graphs with fixed number of vertexes and diameter. Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 6, pp. 74-92. http://geodesic.mathdoc.fr/item/DA_2009_16_6_a6/

[1] Evdokimov A. A., “Lokalno izometricheskie vlozheniya grafov i svoistvo prodolzheniya metriki”, Sib. zhurn. issled. operatsii, 1:1 (1994), 5–12 | MR | Zbl

[2] Emelichev V. A., Melnikov O. I., Sarvanov V. I., Tyshkevich R. I., Lektsii po teorii grafov, Nauka, M., 1990, 383 pp. | MR | Zbl

[3] Konvei D., Sloen N., Upakovki sharov, reshetki i gruppy, T. 1, Mir, M., 1990, 413 pp.

[4] Fedoryaeva T. I., “Raznoobrazie sharov v metricheskikh prostranstvakh derevev”, Diskret. analiz i issled. operatsii. Ser. 1, 12:3 (2005), 74–84 | MR

[5] Fedoryaeva T. I., “Vektory raznoobraziya sharov i svoistva ikh komponent”, Trudy VII Mezhdunarodnoi konferentsii “Diskretnye modeli v teorii upravlyayuschikh sistem” (Moskva, 4–6 marta 2006 g.), Izd-vo MGU, M., 2006, 374–378

[6] Fedoryaeva T. I., “Vektory raznoobraziya sharov dlya grafov i otsenki ikh komponent”, Diskret. analiz i issled. operatsii. Ser. 1, 14:2 (2007), 47–67 | MR

[7] Fedoryaeva T. I., “Tochnye verkhnie otsenki komponent vektorov raznoobraziya sharov dlya grafov s zadannym chislom vershin i diametrom”, Materialy XVII Mezhdunar. shkoly-seminara “Sintez i slozhnost upravlyayuschikh sistem” im. akademika O. B. Lupanova (Novosibirsk, 27 oktyabrya–1 noyabrya 2008 g.), Izd-vo IM SO RAN, Novosibirsk, 2008, 167–172