Monomial graphs and generalized quadrangles

Proof of a conjecture on monomial graphs

“…In [6] and [13], the above conjectures were shown to be true under various additional conditions. The main objective of the present paper is to confirm Conjecture 1.1.…”

“…The results [6] and [13] (see more on them in the next section) suggest that for odd q, monomial graphs of girth at least eight are isomorphic to Γ 3 (q). In fact, the main conjecture of [6] and [13] is the following.…”

“…[6, Theorem 3] and the proof of [6, Theorem 1] allow one to choose p 0 (3) = 5 and p 0 (1) = 3. However, in general, the function p 0 (r) given in [13] is not explicit.…”

“…This motivated papers [6] and [13]. Another reason for looking at the monomial graphs first is the following: For even q, the monomial graphs do lead to a variety of non-isomorphic generalized quadrangles; see Cherowitzo [4], Glynn [9] and [10], Payne [19], and [22].…”

“…In an attempt to prove Conjecture 1.1, two more related conjectures were proposed in [6] and [13]. For an integer 1 ≤ k ≤ q − 1, let…”