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

#### Abstract

For any non-abelian group *G*, the non-commuting graph of *G*, Γ=Γ_{G}, is a graph with vertex set *G* \ *Z*(*G*), where *Z*(*G*) is the set of elements of *G* that commute with every element of *G* and distinct non-central elements *x* and *y* of *G* are joined by an edge if and only if *xy* ≠ *yx*. 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 *M*(*D*_{2n},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 *M*(*G*,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 *G*, the non-commuting graph of the Moufang loop *M*(*G*,2), is 3-split if and only if *G* is isomorphic to a Frobenius group of order 2*n*, *n* 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 *M*(*D _{2n},2).*

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.9

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