Some structural graph properties of the non-commuting graph of a class of finite Moufang loops

• Main Authors: Hamideh Hasanzadeh Bashir, Karim Ahmadidelir
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2020-10-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: loop theory, finite moufang loops, chein loops, non-commuting graph of a finite group, perfect graphs, chordal graphs, split graphs
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/473

<p>For any non-abelian group <em>G</em>, the non-commuting graph of <em>G</em>, Γ=Γ_G, is a graph with vertex set <em>G</em> \ <em>Z</em>(<em>G</em>), where <em>Z</em>(<em>G</em>) is the set of elements of <em>G</em> that commute with every element of <em>G</em> and distinct non-central elements <em>x</em> and <em>y</em> of <em>G</em> are joined by an edge if and only if <em>xy</em> ≠ <em>yx</em>. This graph is connected for a non-abelian finite group and has received some attention in existing literature. Similarly, the non-commuting graph of a finite Moufang loop has been defined by Ahmadidelir. He has shown that this graph is connected (as for groups) and obtained some results related to the non-commuting graph of a finite non-commutative Moufang loop. In this paper, we show that the multiple complete split-like graphs are perfect (but not chordal) and deduce that the non-commuting graph of Chein loops of the form <em>M</em>(<em>D</em>_2<em>n</em>,2) is perfect but not chordal. Then, we show that the non-commuting graph of a non-abelian group G  is split if and only if the non-commuting graph of the Moufang loop <em>M</em>(<em>G</em>,2) is 3-split and then classify all Chein loops that their non-commuting graphs are 3-split. Precisely, we show that for a  non-abelian group <em>G</em>, the non-commuting graph of the Moufang loop <em>M</em>(<em>G</em>,2), is 3-split if and only if <em>G</em> is isomorphic to a Frobenius group of order 2<em>n</em>, <em>n</em> is odd, whose Frobenius kernel is abelian of order n. Finally, we calculate the energy  of  generalized and multiple splite-like graphs, and discuss about the energy and also the number of spanning trees in the case of  the non-commuting graph of  Chein loops of the form <em>M</em>(<em>D_2<em>n</em>,2).</em></p>