Exact values of girth for some graphs D(k,q) and upper bounds of the order of cages

Abstract

Let q be a prime power and k∈{5,7,9,11}. In this paper it is shown that the girth of a graph D(k,q) is equal to k+5. As a consequence, explicit examples of graphs which provide the best known upper bounds of the order of (r,g)-cages, r≥5, g∈{10,14,16}, are given.