Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud’ in their book “Characters of finite groups, Part $1$”, we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character coedgree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.