Limits and Periodicity of Metamour $2$-Distance Graphs
| Autor: | Erickson, William Q., Herden, Daniel, Meddaugh, Jonathan, Sepanski, Mark R., Minyard, Mitchell, Rosengartner, Kyle |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$ obtained by iterating the 2-distance operation. In particular, we classify the connected graphs with period 3, and we partially characterize those with period 2. We then study two families of graphs whose 2-distance sequence is eventually periodic: namely, generalized Petersen graphs and complete $m$-ary trees. For each family, we show that the eventual period is 2, and we determine the pre-period and the two limit graphs of the sequence. Comment: 34 pages, 13 figures |
| Databáze: | arXiv |
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