Bipartite divisor graphs for integer subsets

Abstract

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph $\Ga(X)$ and the prime vertex graph $\Delta (X)$, 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 $\Gamma (X)$ or $\Delta (X)$ to contain a complete subgraph of size 3 or 4.

Cited by in corpus (1)

• The influence of conjugacy class sizes on the structure of finite groups: a survey