Exact values of girth for some graphs \(D\left({k},{q}\right)\) and upper bounds of the order of cages

Abstract

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