The regular models of a logic program are a particular type of partial (3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependence graph of a normal logic program to analyze the existence, unicity and multiplicity of non-trivial regular models for the program. We show three main results: 1) a necessary condition for the existence of non-trivial regular models, 2) a sufficient condition for the unicity of regular models, and 3) two upper bounds for the number of regular models based on positive feedback vertex sets. The first two conditions generalize the existing results obtained by You and Yuan (1994) for well-founded stratification logic programs. The third result is new to the best of our knowledge. Key to our proofs is a connection that we establish between logic programs and Boolean network theory.