On the non-commuting graph in finite Moufang loops

Ahmadidelir, Karim

doi:10.1142/s0219498818500706

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…The commuting graph (and its complement), the intersection graph of cyclic subgroups, and the power graph have also been studied for quasigroups and loops, especially for the classes of Moufang and Bol loops: see, for example, [9,61,62,101]. Moufang loops form a class of loops which is perhaps closest to groups: a Moufang loop is a loop satisfying the identity z(x(zy)) = ((zx)z)y (a weakening of the associative law).…”

Section: Applicationsmentioning

confidence: 99%

Graphs defined on groups

Cameron

2021

Preprint

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These notes concern aspects of various graphs whose vertex set is a group G and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action of the automorphism group of G). The particular graphs I will chiefly discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph.My main concern is not with properties of these graphs individually, but rather with comparisons between them. The graphs mentioned, together with the null and complete graphs, form a hierarchy (as long as G is non-abelian), in the sense that the edge set of any one is contained in that of the next; interesting questions involve when two graphs in the hierarchy are equal, or what properties the difference between them has. I also consider various properties such as universality and forbidden subgraphs, comparing how these properties play out in the different graphs. (There are so many papers even on the power graph that a complete survey is scarcely possible.)I have also included some results on intersection graphs of subgroups of various types, which are often in a "dual" relation to one of the other graphs considered. Another actor is the Gruenberg-Kegel graph, or prime graph, of a group: this very small graph has a surprising influence over various graphs defined on the group. I say little about Cayley graphs, since (except in special cases) these are not invariant under the automorphism group of G.Other graphs which have been proposed, such as the nilpotence, solvability, and Engel graphs, will be touched on rather more briefly. My emphasis is on finite groups but there is a short section on results for infinite groups.Proofs, or proof sketches, of known results have been included where possible. Also, many open questions are stated, in the hope of stimulating further investigation.The graphs I chiefly discuss all have the property that they contain twins, pairs of vertices with the same neighbours (save possibly one another). Being equal or twins is an equivalence relation, and the automorphism group of the graph has a normal subgroup inducing the symmetric group on each equivalence class. For some purposes, we can merge twin vertices and get a smaller graph. Continuing until no further twins occur, the result is unique independent of the reduction, and is the trivial 1-vertex graph if and only if the original graph is a cograph. So I devote a section to cographs and twin reduction, and another to the consequences for automorphism groups. In addition, I discuss the question of deciding, for each type of graph, for which groups is it a cograph. Even for the groups PSL(2, q), this leads to difficult number-theoretic questions.There are briefer discussions of general Aut(G)-invariant graphs, and structures other than groups (such as semigroups and rings).

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Section: Applicationsmentioning

confidence: 99%

Graphs defined on groups

Cameron

2021

Preprint

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“…For instance, one may see [1,10,15,17]. Similarly, the non-commuting graph of a finite Moufang loop has been defined by Ahmadidelir in [3]. He has defined this graph as follows: Let M be a Moufang loop, then the vertex set is M \C(M ) and two vertices x and y joined by an edge whenever [x, y] = 1.…”

Section: Introductionmentioning

confidence: 99%

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

Bashir

Ahmadidelir

2020

EJGTA

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For any non-abelian group G, the non-commuting graph of G, Γ = Γ G , is 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. The non-commuting graph of a finite Moufang loop has been defined by Ahmadidelir. In this paper, we show that the multiple complete split-like graphs and the non-commuting graph of Chein loops of the form M (D 2n , 2) are 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. Precisely, we show that 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 (D 2n , 2).

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“…A quasigroup is a non-empty set with a binary operation such that for every three elements x, y and z of that, the equation xy = z has a unique solution in the set, whenever two of the three elements are specified. A quasi-group with a neutral element is called a loop, and following [5,[1][2][3], and one may see the definition of Moufang loop satisfying four tantamount relators. These loops are of interest because they retain main properties of the groups [4,5].…”

Section: Introductionmentioning

confidence: 99%

Certain numerical results in non-associative structures

Azizi

Doostie

2019

Math Sci

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The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the nth-commutativity degree of a finite algebraic structure S, denoted by P n (S) , is the probability that for chosen randomly two elements x and y of S, the relator x n y = yx n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.

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On the non-commuting graph in finite Moufang loops

Cited by 4 publications

References 22 publications

Graphs defined on groups

Graphs defined on groups

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

Certain numerical results in non-associative structures

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