#### Vol. 8, No. 2, 2015

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Iteration digraphs of a linear function

### Hannah Roberts

Vol. 8 (2015), No. 2, 221–232
##### Abstract

An iteration digraph $G\left(n\right)$ generated by the function $f\left(x\right)\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}n$ is a digraph on the set of vertices $V=\left\{0,1,\dots ,n-1\right\}$ with the directed edge set $E=\left\{\left(v,f\left(v\right)\right)\mid v\in V\right\}$. Focusing specifically on the function $f\left(x\right)=10x\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}n$, we consider the structure of these graphs as it relates to the factors of $n$. The cycle lengths and number of cycles are determined for various sets of integers including powers of 2 and multiples of 3.

##### Keywords
digraph, cycle, congruence
##### Mathematical Subject Classification 2010
Primary: 05C20, 11A07