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Mathematics 2019, 7(4), 312; https://doi.org/10.3390/math7040312

Linear Operators That Preserve the Genus of a Graph

1
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA
2
School of Computational Sciences, Korean Institute for Advanced Study, Seoul 02455, Korea
3
Department of Mathematics, Jeju National University, Jeju 63243, Korea
*
Author to whom correspondence should be addressed.
Received: 6 March 2019 / Revised: 21 March 2019 / Accepted: 25 March 2019 / Published: 28 March 2019
(This article belongs to the Special Issue Algebra and Discrete Mathematics)
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PDF [223 KB, uploaded 28 March 2019]

Abstract

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion. View Full-Text
Keywords: linear operator; genus of a graph; vertex permutation linear operator; genus of a graph; vertex permutation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Beasley, L.B.; Kim, J.H.; Song, S.-Z. Linear Operators That Preserve the Genus of a Graph. Mathematics 2019, 7, 312.

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