On the Genus of the Nilpotent Graphs of Finite Groups |
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Authors: | Ashish Kumar Das Deiborlang Nongsiang |
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Affiliation: | 1. Department of Mathematics , North-Eastern Hill University , Shillong , Meghalaya , India akdasnehu@gmail.com;3. Department of Mathematics , North-Eastern Hill University , Shillong , Meghalaya , India |
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Abstract: | The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal. |
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Keywords: | Finite group Nilpotent graph |
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