  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.
