Limits and Periodicity of Metamour $2$-Distance Graphs
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929485037699072 |
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| author | Erickson, William Q. Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Minyard, Mitchell Rosengartner, Kyle |
| author_facet | Erickson, William Q. Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Minyard, Mitchell Rosengartner, Kyle |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_02306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limits and Periodicity of Metamour $2$-Distance Graphs Erickson, William Q. Herden, Daniel Meddaugh, Jonathan Sepanski, Mark R. Minyard, Mitchell Rosengartner, Kyle Combinatorics Primary: 05C12, 05C76, Secondary: 05C38 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. |
| title | Limits and Periodicity of Metamour $2$-Distance Graphs |
| topic | Combinatorics Primary: 05C12, 05C76, Secondary: 05C38 |
| url | https://arxiv.org/abs/2409.02306 |