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

For any non-abelian group G, the non-commuting graph of G, Γ = ΓG, 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 6= 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(D2n, 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(D2n, 2).


Introduction
Let Q be a set with one binary operation. Then it is a quasigroup if the equation xy = z has a unique solution in Q whenever two of the three elements x, y, z ∈ Q are specified. A quasigroup Q is a loop if Q possesses a neutral element e, i.e., if ex = xe = x holds for every x ∈ Q. Moufang loops are loops in which any of the (equivalent) Moufang identities, ((xy)x)z = x(y(xz)), (M 1) x(y(zy)) = ((xy)z)y, (M 2) (xy)(zx) = x((yz)x), (M 3) (xy)(zx) = (x(yz))x. (M 4) holds for every x, y, z ∈ Q. Commutator of x, y and the associator of x, y and z are defined by [x, y] = x −1 y −1 xy and [x, y, z] = ((xy)z) −1 (x(yz)), respectively. We define the commutant (or Moufang center) C(Q) of Q as {x ∈ Q | xy = yx, ∀y ∈ Q}. The center Z(Q) of a Moufang loop Q is the set of all elements of Q which commute and associate with all other elements of Q. A non-empty subset P of Q is called a subloop of Q if P is itself a loop under the binary operation of Q, in particular, if this operation is associative on P , then it is a subgroup of Q. A subloop N of a loop Q is said to be normal in Q if xN = N x; x(yN ) = (xy)N ; N (xy) = (N x)y; for every x, y ∈ Q. In Moufang loop Q, the subloops Z(Q) and C(Q) are normal subloops. For more details about the Moufang loops one may see [8,16,13]. In 1974, Chein introduced a class of non-associative Moufang loops M (G, 2), so called Chein loops. For a group G and a new element u, (u / ∈ G), M (G, 2) = G ∪ Gu such that the multiplication with the new binary operation • is defined as follows:        g • h = gh, g, h ∈ G, g • (hu) = (hg)u, g ∈ G, hu ∈ Gu, (gu) • h = (gh −1 )u, gu ∈ Gu, h ∈ G, (gu) • (hu) = h −1 g, gu, hu ∈ Gu.
Clearly, the Moufang loop M (G, 2) is non-associative if and only if G is non-abelian, see [8]. In [2], Ahmadidelir has investigated some probabilistic properties of M (G, 2), such as its commutativity degree.
There are many papers on assigning a graph to a ring or a group in order to investigation of their algebraic properties. 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 distinct non-central elements x and y of G are joined by an edge if and only if xy = yx. This graph is connected with diameter 2 and girth 3 for a non-abelian finite group and has received some attention in existing literature. 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. 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.
We will denote a complete graph with n vertices by K n . All graphs considered in this paper are finite and simple. For a graph Γ, we denote its vertex and edge sets by V (Γ) and E(Γ), respectively. www.ejgta.org The complement of Γ is denoted byΓ. A graph Γ = (V, E), is called k−partite where k > 1, if it is possible to partition V into k subsets V 1 , V 2 , . . . , V k , such that every edge of E joins a vertex of V i to a vertex of V j , i = j. A clique in a graph Γ is an induced subgraph whose all vertices are pairwise adjacent. The maximum size of a clique in a graph Γ is called the clique number of Γ and denoted by ω(Γ). A subset X of the vertices of Γ is called an independent set (or stable) if the induced subgraph on X has no edges. The maximum size of an independent set in a graph Γ is called the independence number of Γ and denoted by α(Γ). The vertex chromatic number of a graph Γ is denoted by χ(Γ), and it is the minimum k for which k−vertex coloring of a graph Γ such that no two adjacent vertices have the same color. For a subset S of V (Γ), N Γ [S] is the set of vertices in Γ which are in S or adjacent to a vertex in S. If N Γ [S] = V (Γ) then S is said to be a dominating set of the vertices in Γ. The minimum size of a dominating set of the vertices in Γ is dominating number of Γ and denoted by γ(Γ). A vertex cover of a graph Γ is a set Q ⊆ V (Γ) such that contains at least one endpoint of every edge. The minimum size of a vertex cover is denoted by β(Γ). Our other used notations about graphs are standard and for more details one may see [6,7,11].
There is a relation between α(Γ) and β(Γ) as follows: A perfect graph Γ, is a graph in which for every induced subgraph its clique number is equal to its chromatic number. A graph Γ is called weakly perfect graph if ω(Γ) = χ(Γ). So, all perfect graphs are weakly perfect. A chordal graph is one in which all cycles of order four or more have a chord, which is an edge that is not part of cycle but connects two vertices of the cycle. The class of Chordal graphs is a subset of the class of perfect graphs. For more information about these types of graphs, one may see [12,14]. We have the following Theorem about perfect graphs, called strongly perfect graph theorem, or Berg Theorem.
A graph is called k-regular, if the vertices of the graph are of the same degree k and a strongly regular graph S with parameters (n, k, λ, µ) is a k−regular graph of order n such that each pair of adjacent vertices has λ common neighbors and each pair of non-adjacent vertices has in which µ common neighbors. Let Γ 1 = (V 1 , E 1 ) and Γ 2 = (V 2 , E 2 ) be undirected simple graphs. The union The complete product Γ 1 ∇Γ 2 of graph Γ 1 and Γ 2 is a graph obtained from Γ 1 ∪ Γ 2 by joining every vertex of Γ 1 to every vertex of Γ 2 . For every a, b, n ∈ N , a complete split, or simply, a split graph, is the graphK a ∇K b and denoted by CS a b . By a theorem of Földes and Hammer ( [12], Theorem 6.3), a graph is (complete) split iff contains no induced subgraph isomorphic to 2K2, C 4 or C 5 . Also, an undirected graph is split if and only if its complement is split ( [12], Theorem 6.1). Clearly, every split graph is chordal and so perfect, but the converses are not true. More generally, a multiple complete split-like graph isK a ∇(nK b ) and denoted by M CS a b,n . Specially, in this paper, for n = 3 we call M CS a b,3 as a 3−split graph. We generalize the above definitions as follows: www.ejgta.org Definition 1.1. The generalized complete split-like graph is GCS a k =K a ∇S such that S is a strongly regular graph with parameters (n, k, λ, µ). The multiple generalized complete split-like graph is GM CS a k,m =K a ∇(mS).
The laplacian matrix of a simple graph Γ with n vertices, is defined as L(Γ) = D(Γ) − A(Γ), where A(Γ) is its adjacency matrix and D(Γ) = (d 1 , . . . , d n ) is the diagonal matrix of the vertex degrees in Γ. For any graph Γ, the energy of Γ is defined as ξ(Γ) = n i=1 |λ i |, where λ 1 , . . . , λ n are the eigenvalues of the adjacency matrix of Γ. A spanning tree of a graph Γ is an induced subgraph of Γ, which is a tree and contains every vertex of Γ.
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 nonabelian 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 (D 2n , 2). We recall the following Proposition and Theorems in order to provide some tools to these purposes. can be calculated by the following formula: 14], Theorem 1) For i = 1, 2, let Γ i be r i −regular graphs with n i vertices. Then the characteristic polynomial of the complete product of these two graphs is as follows:

Some basic graph properties of the Moufang loop M (D 2n , 2)
Let D 2n denote the dihedral group of order 2n, which has the following presentation: D 2n = a, b| a n = b 2 = (ab) 2 = 1 .
In this section, we want to study the non-commuting graph of the Moufang loops M (D 2n , 2), simply denoted by Γ. We will use the following Lemma in next sections.
The following Lemma determines the structure of the non-commuting graph of the Moufang loop M = M (D 2n , 2).
(a) If n is odd then Γ M ∼ =K n−1 ∇S, such that S is a strongly regular graph with parameters (3n, n − 1, n − 2, 0).
For every 0 ≤ i, j ≤ n − 1, since a i a j = a j a i , t 1 is an independent set and from the relations • a i and a i • (a j bu) = (a j bu) • a i , we find that all vertices of t 1 are adjacent to all vertices of each of the sets t 2 , t 3 and t 4 . Also, by the relations Similarly, we can show that the induced subgraph [t 3 ] and [t 4 ] of Γ, are cliques. Hence, Γ ∼ =K n−1 ∇3K n and the graph Γ is 3−split and 3K n ∼ = S, where S is a strongly regular graph with parameters (3n, n − 1, n − 2, 0). b) Let n be an even integer. Again, we can partition the vertices of Γ into four sets, as follows: . . , a n 2 −1 , a n Since each pair of elements of t 1 commute, so the induced subgraph [t 1 ] is an independent set, that means [t 1 ] ∼ =K n−2 . Also, every element in M commutes with its inverse and since, ∀x ∈ t i , (i = 2, 3, 4), its inverse x −1 belongs to t i . Therefore, every element of t i , (i = 2, 3, 4) is adjacent to each vertex in t i , i = 2, 3, 4, except its inverse. Also any two elements x ,y in t i , (i = 2, 3, 4) commute if and only if |i − j| = n 2 , where x = a i u or a i b, a i bu and y = a j u or a j b, a j bu. Then [t i ] ∼ = S , where S is a strongly regular graph with parameters (n, n − 2, n − 3, n − 2). Finally, for every 2 ≤ i, j ≤ 4 there is no edge of Γ such that joins a vertex of t i to a vertex of t j , i = j, but each vertex in t 1 joins to each vertex in t i , (i = 2, 3, 4). Therefore, Γ M ∼ =K n−2 ∇3S.
In the following Theorem, we derive some important graph properties of Γ M (D 2n ,2) . www.ejgta.org (a) If n is odd then: (b) If n is even then: Proof. a) By Lemma 2.1, the non-commuting graph of M (D 2n , 2) is a generalized complete splitlike graph for any odd integer n. Then Γ =K n−1 ∇S in which S is a strongly regular graph with parameters (3n, n − 1, n − 2, 0), where V (K n−1 ) = {a, a 2 , . . . , a n−1 } and S ∼ = 3K n . So this graph is 3−split. By the structure of Γ, since every vertex of each copy of K n is joined to every vertex of is the largest clique in Γ. So, ω(Γ) = n + 1. We need n distinct colors for coloring any K n and only one color for coloringK n−1 which is distinct with the previous ones. So, χ(Γ) = n + 1. The set of vertices ofK n−1 is the largest independent set, so α(Γ) = n − 1. By Lemma 1.1, we have β(Γ) = 4n−1−(n−1) = 3n. Clearly, the set of vertices of 3K n has the minimum size of a vertex cover. Any vertex ofK n−1 is dominating all vertices of S, and any vertex of S is dominating all vertices inK n−1 . Thus γ(Γ) = 2.
In order to find the clique number, we may choose one vertex ofK n−2 and the other vertices from only one copy of S's. By definition, every vertex is not joined to its inverse, so, we can choose n 2 vertices of S and hence, ω(Γ) = n 2 + 1. The color of every vertex in S is co-color with its inverse. Therefore, the chromatic number of S is equal to n 2 , and so the maximum color number for all the vertices of 3S is equal to n 2 . By only one color distinct from n 2 −color in 3S, we can colorK n−2 . So, χ(Γ) = n 2 + 1. For n = 6,K n−2 have four independent vertices, but with two non-adjacent vertices chosen from any of the copies of S, we get 6 independent vertices. Therefore, in this case α(Γ) = 6. Now, for n ≥ 8, the setK n−2 is the largest independent set and so, α(Γ) = n − 2. By using Lemma 1.1, we have β(Γ) = n(Γ) − α(Γ). Hence, if n = 6 then β(Γ) = 16, else if n ≥ 8 then β(Γ) = 4n − 2 − (n − 2) = 3n. By choosing any vertex inK n−2 and the other in one of the copies of S, the domination set of Γ will be determined. Hence, γ(Γ) = 2.

About perfectness and splitness of the non-commuting graph of a Moufang loop
In this section, first we show that the multiple complete split-like graphs are perfect and then characterize all Chein loops that their non-commuting graphs are 3−split-like. Proof. Let Γ ∼ =K a ∇(nK b ) and C be an odd cycle. If all vertices of C lie in only one copy of K b 's, clearly this cycle has a chord. Also, if some vertices of C lie in more than one copy of K b 's, then since in this case C has some vertices ofK a and also these vertices inK a are adjacent to each vertex of K b , therefore, the cycle has a chord. In addition, the complement graph,Γ, is a disconnected graph of the formΓ ∼ = K a ∪ S such that S is strongly regular graph with parameters (nb, (n − 1)b, (n − 2)b, (n − 1)b) or S ∼ = T nb,b , which is a complete n−partite graph with nb vertices, and hence, each part has b vertices. Clearly, any cycle in K a has a chord. If C be an odd cycle in S, then by structure of S, there is an intersection of C with more than three sections of S and these vertices are adjacent to any of the vertices in other sections and so, C has a chord. If C has an instruction with only two sections of S, then the induced subgraph of these sections will be a bipartite graph such that there is no any odd cycle in it. Now, by Berg Theorem ([9], Theorem 1.2) Γ is a perfect graph. Let Γ ∼ =K a ∇(nK b ) and x 1 , x 2 ∈K a , x 1 = x 2 . Take x 3 and x 4 from two distinct copies of K b 's. Now the induced subgraph of Γ generated by x 1 , x 2 , x 3 and x 4 is a cycle of length four without a chord. So, by definition, Γ is not chordal.
Similar to the proof of the first part, CS a b,n ∼ =K a ∇K b is perfect, but there is no cycle of length four or more without any chord and so this is a chordal graph. This completes the proof. Proof. Let Γ = Γ(M (D 2n , 2)), where n be an odd integer. Then by Lemma 2.1 (a), Γ ∼ =K n−1 ∇(3K n ) and by Theorem 3.1, Γ is perfect but not chordal. If n be an even integer then by Lemma 2.1(b), Γ ∼ =K n−2 ∇3S such that S is a strongly regular graph with parameters (n, n − 2, n − 3, n − 2). Assume that C is an odd cycle in Γ with length 5 or more, the length of the longest cycle without chord in each copy of S is equal to 4. Then there are some vertices ofK n−2 in C, and these vertices are adjacent to each vertex in 3S. Therefore, C have a chord. On the other hand, Γ ∼ = K n−2 ∪ ( n 2 K 2 ∇ n 2 K 2 ∇ n 2 K 2 ). Let C be a cycle in Γ. Clearly, every cycle in K n−2 have a chord and if C be an odd cycle in n 2 K 2 ∇ n 2 K 2 ∇ n 2 K 2 , then C have an intersection with more than two parts of n 2 K 2 , where one of them have more than one vertex in C, and these vertices adjacent to all vertices of C in other parts and so, C have a chord and by Theorem ([9], Theorem 1.2), Γ is perfect. The induced subgraph consist of any two vertices ofK n−2 and two non-adjacent vertices of S is a cycle with length 4 without chord then Γ is not chordal.
Remark 3.1. The generalized multiple complete split-like graph GM CS a k is not perfect. As a counterexample, let we have a generalized complete split-like graph Γ ∼ =K a ∇(nS) in which S is a Peterson graph. This graph is not perfect, since it has a cycle of length 5 without any chord. Recall that a Peterson graph is a strongly regular graph with parameters (10, 3, 0, 1). Proof. Let Γ M be 3−split of the form Γ M = I∇3C, where I is an independent set and C is a complete graph. First we show that Z(G) = C(M ). By Lemma( [3], Lemma 3.10), C(M ) ⊆ Z(G). Let Z(G) C(M ). Then there exists x ∈ Z(G) such that x / ∈ C(M ). Also, there exists yu ∈ Gu, where x • (yu) = (yu) • x, which yields (yx)u = (yx −1 )u. Therefore, x = x −1 and x ∈ I. So, every vertex y in each copy of C is adjacent to x and so xy = yx. But x ∈ Z(G) then for every g ∈ G, we have xg = gx. Hence G ⊆ I. Now, let g ∈ G \ Z(G). So, there exist t ∈ G such that tg = gt but in this case t, g ∈ I and this is a contradiction, since I is an independent set. So, G = Z(G) and this contradicts with non-abelianity of G. Thus Z(G) = C(M ). Obviously, every element of 3C is an involution. Let x ∈ 3C and x = x −1 . So, since each element of Gu has order 2 then x ∈ G. Put 3C = C 1 ∪ C 2 ∪ C 3 , where each C i is equal to a copy of C, (1 ≤ i ≤ 3). Without loss of generality, let x ∈ C 1 and x −1 ∈ C 2 (note that xx −1 = x −1 x). Let y ∈ G\Z(G) and y / ∈ x . Then since every element of G which commutes with x, also commutes with x −1 , so if y ∈ C 1 then xy = yx, and therefore x −1 y = yx −1 , but x −1 ∈ C 2 and this is a contradiction. Similarly, the case y ∈ C 2 will reach to a contradiction. So, y ∈ I or y ∈ C 3 . Now, consider the element xy. By the same reason as above, we have xy ∈ I or xy ∈ C 3 . Trivially, xy = x, x −1 . We have four cases as below: Case 1. Let y, xy ∈ I. Then y(xy) = (xy)y ⇒ yx = xy. which is a contradiction, since y is adjacent to every element of C 1 .
Case 2. Let y ∈ I and xy ∈ C 3 . Then x ∈ C 1 ⇒ x(xy) = (xy)x, (x, y ∈ G) ⇒ xy = yx and we have the same contradiction as in case 1.

Case 3.
Let y ∈ C 3 and xy ∈ I. Then (xy)y = y(xy) ⇒ xy = yx, which is also a contradiction since y ∈ C 3 and x ∈ C 1 . Case 4. Let y, xy ∈ C 3 . Then we have y(xy) = (xy)y ⇒ xy = yx and we obtain a similar contradiction as in case 3.
Therefore, every element of 3C has order 2. On the other hand, Γ G is always connected and it is the induced subgraph of Γ M . Therefore, Γ G ∼ = K m , (K m ⊆ C) or Γ G ∼ = I ∇nC such that I ⊆ I, C ⊆ C and nC = ∪ n i=1 C i , where 1 ≤ n ≤ 3, and each C i is a subset of one copy of C's. If Γ G ∼ = K m , then the order of every element of G will be equal to 2, so G must be abelian, which is absurd. Therefore, we get, Γ G ∼ = I ∇nC . If n = 1 then Γ G is split. Suppose that 1 = x, y ∈ G, x ∈ C 1 and y ∈ C 2 , then xy = yx and there exists z ∈ I where yz = zy and xz = zx. So, xy ∈ G. If xy ∈ I , then x(xy) = (xy)x and so, x 2 y = x(yx). Therefore, x 2 y = x(xy) and this is a contradiction. If xy ∈ C 1 then x(yx) = (xy)x and x 2 y = x 2 y, and it is a contradiction, and if xy ∈ C 2 then y(xy) = (xy)y and y 2 x = y 2 x, and it is also a contradiction. Finally, let xy ∈ C 3 . Now, xu ∈ M (G, 2) then: 1) If xu ∈ I or xu ∈ C 1 , then (xu) • x = x • (xu) and so (xx −1 )u = (xx)u. Therefore, u = x 2 u, this is a contradiction. So, every element of C in Γ M is of order 2 therefore, x 2 = 1.
Therefore, Γ G ∼ = I ∇C and Γ G is split.
Conversely, let Γ G be split. Then Γ G ∼ = I∇C. We show that Γ M is 3−split. By splitness of Γ G and Lemmas (  Suppose that there exist two vertices b i u and b j u such that are not adjacent. So,

each element of C is an involution and which yields to a contradiction.
Claim 3. There is no edge between V (Iu) and V (Cu).
Suppose that there exist two vertices a i u and b j u such that (a i u) On the other hand b j a i ∈ G. So, b j a i ∈ I or b j a i ∈ C. 1) If b j a i ∈ I then (b j a i )a i = a i (b j a i ) and b j a i = a i b j , which yields to a contradiction.
2) If b j a i ∈ C then (b j a i ) 2 = 1 and this is a contradiction. Therefore, any two elements of V (Iu) and V (Cu) are non-adjacent.

Claim 4.
There is no edge between V (C) and V (Cu).

Suppose that there exist two vertices
and this is a contradiction. Therefore any two elements of V (C) and V (Cu) are non-adjacent.

Claim 5.
There is no edge between V (C) and V (Iu).
Suppose that there exist two vertices b i and a j u such that b i • (a j u) = (a j u) • b i . Then (a j b i )u = (a j b −1 i )u and a j b i = a j b i . This is a contradiction. Therefore, any two vertices in V (C) and V (Iu) are non-adjacent.
Claim 6. Every vertex in V (Iu) is adjacent to every vertex in V (I).
Suppose that there exist two vertices a i and a j u such that a i • (a j u) = (a j u) • a i . Then (a j a i )u = (a j a −1 i )u and a j a i = a j a −1 i . So, a i = a −1 i . Therefore, a 2 i = 1 and this is a contradiction.
Claim 7. Every vertex in V (Cu) is adjacent to every vertex in V (I).
Suppose that there exist two vertices a i ∈ I and b j u ∈ Cu such that , 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.

About the energy and the number of spanning trees of generalized and multiple splite-like graphs
In this section, we are going to calculate the energy of generalized complete and multiple splite-like graphs and derive 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). Proof. Let P K b (λ) be the characteristic polynomial of K b . Then, and PK a (λ) = (−λ) a .
Finally, in the following Theorems, we count the number of spanning trees of the non-commuting graph Γ M , where M = M (D 2n , 2), for odd and even n, separately, and they lead us to an important result.
By Theorem ( [5], Theorem 4.11), we have κ = det(L+J) (4n−1) 2 . Therefore,  Proof. There are 4n − 2 vertices in this graph and they are in t 1 , t 2 , t 3 , t 4 . Each of t i , 2 ≤ i ≤ 4, have n vertices of degree 2n − 4 and t 1 have n − 2 vertices of degree 3n. By the structure of the graph Γ in 2.1, the matrix of the vertex degree namely D, of this graph is: www.ejgta.org Some structural graph properties of the non-commuting graph ... | H. H. Bashir and K. Ahmadidelir and the adjacent matrix of the graph has the form: By Lemma 2.1, each vertex in every t i (2 ≤ i ≤ 4), is connected to the other vertices except its inverse element and itself, and so, such that I and J are square matrices of order n 2 in X. So, Hence, We have in which the order of I is equal to n 2 . Now we obtain where C = 3nI n−2 + J n−2 and Therefore, det C = 2(3n) n−3 (2n − 1) and by using Theorem 1.1, we have Also, from relations 12, 13 and 17, we obtain det B = 2 3n−2 (n − 1) and from replacing 19 in κ = det(L+J) (4n−2) 2 , we get κ(Γ M ) = 2 3n−3 (3n) n−3 (n − 1)  Proof. By Example 1 in [4], the non-commuting graph of G = D 2n , when in is odd, is a split graph and Γ G ∼ = I∇C, where I is an independent set with n − 1 vertices and C ∼ = K n . So, the degree matrix of Γ G has the form: Therefore, κ(Γ G ) = det(L + J) (2n − 1) 2 = (2n − 1) n−1 n n−2 .

Conclusion
In this research work, we studied some properties of the non-commuting graph of a class of finite Moufang loops. Also, we proved that the multiple complete-like graphs and the noncommuting graph of Chein loops of the form M (D 2n , 2) are perfect, and both graphs are non chordal. Finally, we characterized when the non-commuting graph of Moufang loop M (G, 2) is 3-splite and we give the energy of generalized and multiple splite-like graphs. In future, we will try to study the similar graph properties of the non-commuting graph for the simple Moufang loops and characterize relations between any group G with the non-commuting graph M (G, 2).