Linear Operators That Preserve the Genus of a Graph

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...

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Bibliographic Details
Published in:Mathematics (Basel) 2019-04, Vol.7 (4), p.312
Main Authors: Beasley, LeRoy B., Kim, Jeong Han, Song, Seok-Zun
Format: Article
Language:English
Subjects:
Apexes
Food science
genus of a graph
Graphs
linear operator
Linear operators
Permutations
vertex permutation
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math7040312