| Summary: | If a graph can be embedded in a smooth orientable surface of genus <i>g</i> without edge crossings and can not be embedded on one of genus <inline-formula> <math display="inline"> <semantics> <mrow> <mi>g</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> without edge crossings, then we say that the graph has genus <i>g</i>. We consider a mapping on the set of graphs with <i>m</i> vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with <i>m</i> vertices, we consider and investigate those linear operators which map graphs of genus <i>g</i> to graphs of genus <i>g</i> and graphs of genus <inline-formula> <math display="inline"> <semantics> <mrow> <mi>g</mi> <mo>+</mo> <mi>j</mi> </mrow> </semantics> </math> </inline-formula> to graphs of genus <inline-formula> <math display="inline"> <semantics> <mrow> <mi>g</mi> <mo>+</mo> <mi>j</mi> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>j</mi> <mo>≤</mo> <mi>g</mi> </mrow> </semantics> </math> </inline-formula> and <i>m</i> sufficiently large. We show that such linear operators are necessarily vertex permutations.
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