2009/10/28 by Mohammad A. Iranmanesh, Cheryl E. Praeger, Iranmanesh, Mohammad A. +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.0910.5396
openalex publication_date 2009/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph \Ga(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.