Counting triangles in regular graphs

Abstract

In this paper, we investigate the minimum number of triangles, denoted by , in ‐vertex ‐regular graphs, where is an odd integer and is an even integer. The well‐known Andrásfai–Erdős–Sós Theorem has established that if . In a striking work, Lo has provided the exact value of for sufficiently large , given that . Here, we bridge the gap between the aforementioned results by determining the precise value of in the entire range . This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large .