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Abstract: Inspired by connections described in a recent paper by Mark L. Lewis, betweenthe common divisor graph $\GaX$ and the prime vertex graph $\DeltaX$, for aset $X$ of positive integers, we define the bipartite divisor graph $BX$, andshow that many of these connections flow naturally from properties of $BX$.In particular we establish links between parameters of these three graphs, suchas number and diameter of components, and we characterise bipartite graphs thatcan arise as $BX$ for some $X$. Also we obtain necessary and sufficientconditions, in terms of subconfigurations of $BX$, for one $\GammaX$ or$\DeltaX$ to contain a complete subgraph of size 3 or 4.

Autor: Mohammad A. Iranmanesh, Cheryl E. Praeger


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