2014/09/22 by Maximilien Gadouleau, Gadouleau, Maximilien, Adrien Richard +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Cellular Automata and Applications #DNA and Biological Computing #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Information Theory (cs.IT) #cs.DM #cs.IT #math.DS #math.IT
paper · pdf · doi:10.48550/arxiv.1409.6144
arxiv created 2014/09/22 · openalex publication_date 2014/09/22 · arxiv updated 2014/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are interested in the number of fixed points of functions f:An→ An over a finite alphabet A defined on a given signed digraph D. We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on D. We then discover relationships between the number of fixed points of f and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behaviour of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points.