On regular triangle-distinct graphs

The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024) 113695] recently asked whether there exists a regular graph that is triangle-distinct. Here we first showcase the examples of regular, triangle-distinct graphs, and then show that for every natural number k there exists a family of \(2^k\) $2^k$  regular triangle-distinct graphs, all having the same order and size.