M D P I AG
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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.
Beasley, L.B.; Kim, J.H.; Song, S.-Z. Linear Operators That Preserve the Genus of a Graph. Mathematics 2019, 7, 312.