Some structural graph properties of the non-commuting graph of a class of finite Moufang loops
| Main Authors: | Bashir, Hamideh Hasanzadeh; Department of Mathematics, Science and Research Branch, Islamic Azad University Iran, Ahmadidelir, Karim; Department of Mathematics Tabriz Branch, Islamic Azad University Tabriz, Iran |
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| Format: | Article info application/pdf eJournal |
| Bahasa: | eng |
| Terbitan: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB
, 2020
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| Subjects: | |
| Online Access: |
https://www.ejgta.org/index.php/ejgta/article/view/473 https://www.ejgta.org/index.php/ejgta/article/view/473/pdf_144 |
Daftar Isi:
- For any non-abelian group G, the non-commuting graph of G, Γ=ΓG, 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(D2n,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 2n, 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(D2n,2).