Counting triangles in regular graphs

In this paper, we investigate the minimum number of triangles, denoted by t(n, k), in n-vertex k-regular graphs, where n is an odd integer and k is an even integer. The well-known Andrasfai-Erdos-Sos Theorem has established that t (n, k) > 0 if k > 2n/5. In a striking work, Lo has provided the exact value of t (n, k) for sufficiently large n, given that 2n/5 + 12 root n/5 < k < n/2. Here, we bridge the gap between the aforementioned results by determining the precise value of t (n, k) in the entire range 2n/5 < k < n/2. This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large n.