Counting Girth Cycles in the Graphs D(4,q)

A girth cycle refers to a cycle that has the minimum length within a graph. The graphs D(k,q) form an important family of algebraically defined bipartite graphs over finite fields, and their short-cycle structure is closely related to questions in extremal graph theory and finite geometry. Motivated by the problem of determining the edge-girth-regular parameter of D(k,q), we determine the exact number of girth cycles in D(4,q) for every prime power q. Our proof uses a convenient isomorphic model Γ(4,q) and edge-transitivity to reduce the global enumeration to counting the girth cycles containing one fixed edge. Specifically, we prove that when q is odd with q>3, the number of girth cycles in D(4,q) is q5(q−1)2(q−3)/8. Moreover, when q is even, the number of girth cycles in D(4,q) is q5(q−1)2(2q−3)/8. When q=3, the girth is 12 and the number of girth cycles is 729. Together, these results resolve the case k=4 of the problem posed in our earlier work.