EN|RU Volume 16, No 6, 2009, P. 74-92 UDC 519.17 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 Abstract: 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. Fedoryaeva Tat’yana Ivanovna 1 1. S. L. Sobolev Institute of Mathematics, SB RAS, 4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia e-mail: stdd@academ.org © Sobolev Institute of Mathematics, 2015