The maximum number of triangles in regular graphs

We determine the maximum number of triangles in n-vertex k-regular graphs and characterize all extremal graphs for all admissible orders when k is odd and for an explicit range of orders when k is even. For even k, we also give explicit lower and upper bounds for the remaining orders. As a consequence, we obtain a lower bound on the number of independent sets of size three in k-regular graphs and characterize all equality cases. Our exact results confirm the conjecture of Petingi and Rodríguez and the triangle case of a conjecture of Cambie, de Joannis de Verclos and Kang for all odd k and the specified orders when k is even. Moreover, for every even k≥18, we construct counterexamples to both conjectures and obtain a recursive characterization of all extremal graphs of order at least $Ck$, where C is an absolute constant.