On the diameter of Wenger graphs

Let q be a prime power, _q the field of q elements, and n≥1 a positive integer. The Wenger graph W _n (q) is defined as follows: the vertex set of W _n (q) is the union of two copies P and L of (n+1)-dimensional vector spaces over script F sign_q, with two vertices (p ₁,p ₂,p _n+1) P and [l ₁,l ₂,l _n+1] L being adjacent if and only if l _i +p _i =p ₁ l _i-1 for 2≤i≤n+1. Graphs W _n (q) have several interesting properties. In particular, it is known that when connected, their diameter is at most 2n+2. In this note we prove that the diameter of connected Wenger graphs is 2n+2 under the assumption that 1≤n≤q-1.