A new graph based on the semi-direct product of some monoids

Tam metin

(1)Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. RESEARCH. Open Access. A new graph based on the semi-direct product of some monoids Eylem Guzel Karpuz1* , Kinkar Chandra Das2 , Ismail Naci Cangul3 and Ahmet Sinan Çevik4 *. Correspondence: eylem.guzel@kmu.edu.tr 1 Department of Mathematics, Kamil Özdag Science Faculty, Karamanoglu Mehmetbey University, Yunus Emre Campus, Karaman, 70100, Turkey Full list of author information is available at the end of the article. Abstract In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM ). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. MSC: 05C10; 05C12; 05C25; 20E22; 20M05 Keywords: graphs; semi-direct product; monoid presentation. 1 Introduction and preliminaries In this paper, we mainly investigate the interplay between the semi-direct product over monoids and the graph-theoretic properties of the semi-direct product in terms of its relations. In detailed, let us consider a free abelian monoid Fn of rank n and also consider a finite cyclic monoid C. Then, by [], we can define the semi-direct product of Fn by C. Moreover, one can also define a new graph associated with this semi-direct product (see Section . below). Thus the idea in here is to present the interplay between the algebraic semigroup and graph-theoretic properties of this new graph. In fact, by the graphtheoretic properties, we will be interested in the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of the corresponding new graph. In the literature, there are some important graph varieties and works that are related to algebraic and topological structures, namely, Cayley graphs [–] or zero-divisor graphs [–]. But the graph constructed in here is different from those in the previous studies and is also interesting in terms of using algebraic semi-direct products during the construction of the vertex and edge sets. So, this kind of graph not only provides the classification of algebras (monoids, semigroups), but also solves the problems of normal forms of elements, word problem, rewriting system, embedding theorems, extensions of groups and algebras, growth function, Hilbert series, etc. As is well known, these problems are really important in fixed point results since they have a direct connection to nature sciences. First of all, let us recall the semi-direct product of any two monoids and its presentation. Thus, let us take two arbitrary monoids A and K with associated presentations PA = [X; r] and PK = [Y ; s], respectively. Also, let M = K θ A be the corresponding semi-direct prod© 2013 Karpuz et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited..

(2) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Page 2 of 8. uct of these two monoids, where θ is a monoid homomorphism from A to End(K). We note that the reader can find some examples of monoid endomorphisms in []. The elements of M can be regarded as ordered pairs (a, k), where a ∈ A, k ∈ K with multiplication given by (a, k)(a , k ) = (aa , (kθa )k ). The monoids A and K are identified with the submonoids of M having elements (a, ) and (, k), respectively. Furthermore, one can define a standard presentation for M as follows: For every x ∈ X and y ∈ Y , choose a word, which we denote by yθx , on Y such that [yθx ] = [y]θ[x] as an element of K . To establish notation, let us denote the relation yx = x(yθx ) on X ∪ Y by Tyx and write t for the set of relations Tyx . Then. PM = [Y , X; s, r, t] is a standard monoid presentation for the semi-direct product M. We may refer to [, ] for more detailed knowledge about the definition and a standard presentation for the semidirect product of two monoids. Let Fn and C be a free abelian monoid of rank n and a finite cyclic monoid with the presentations PFn = y , y , . . . , yn ; yi yj = yj yi ( ≤ i < j ≤ n) and PC = x; xk = xl ( ≤ l < k) , respectively. By [], if one considers the matrix ⎡. α ⎢α ⎢  M=⎢ ⎢ .. ⎣ . αn. α α. ··· ··· .. .. αn. ···. ⎤ αn αn ⎥ ⎥ .. ⎥ ⎥ . ⎦ αnn. and assumes Mk ≡ Ml (mod d), where d | (k – l), then there exits a semi-direct product M = Fn θ C with the presentation PM = y , y , . . . , yn , x; yi yj = yj yi ( ≤ i < j ≤ n), xk = xl , y x = xyα  yα  · · · yαnn , y x = xyα  yα  · · · yαnn , . . . , α α yn x = xy n y n · · · yαnnn .. (). 1.1 A new graph based on semi-direct products In the following, we define an undirected graph (PM ) = (V , E) associated with PM given in (). Actually, all the results presented in this paper are based on this graph. The vertex set V consists of the following: • all generators yi ( ≤ i ≤ n) and x of PM , α α α • words of the form y i y i · · · ynin ( ≤ i ≤ n) in the presentation (), • words of the form yi yi+ ( ≤ i ≤ n – ) in (). (Here, we omitted the words of the remaining format. In other words, we do not take yi yi+ , yi yi+ , etc. as a vertex in our set),.

(3) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Page 3 of 8. • word of the form xk if l =  in the presentation (). Otherwise, i.e., if l =  then we have a relator xk = x; in that case, since x is in the vertex set, there is no need to take xk as a vertex. The edge E consists of the following: • connect each vertex yi to single x for all  ≤ i ≤ n, • connect each of the adjacent vertices yi and yi+ for all  ≤ i ≤ n – , α α α • connect each vertex yi to the related vertices y i y i · · · ynin for all  ≤ i ≤ n, • connect each of the vertices yi and yi+ to the vertex yi yi+ from both sides for all  ≤ i ≤ n – , α α α • connect the unique vertex x to each vertex of the form y i y i · · · ynin for all  ≤ i ≤ n. Remark  (a) In the construction of the semi-direct product, we assume that all rows of the matrix M are different from each other. This affects our matching in the graph as all α α α vertices y i y i · · · ynin ’s are distinct. (b) To simplify, let us label the vertex yα  yα  · · · yαnn α α y n y n. · · · yαnnn. by I,. yα  yα  · · · yαnn. by II,. ··· ,. by N.. (c) As seen in Figure , the number of vertex and edge sets depends on the number of generators of the free abelian monoid of rank n. Thus we have. .

(4).

(5). V (PM ) = n +  and E (PM ) = n –  if l =  in (),. .

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(7). V (PM ) = n and E (PM ) = n –  if l =  in (). Thus we obtain the graph (PM ) as drawn in Figure .. Figure 1 The graph (PM )..

(8) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Page 4 of 8. 2 Graph theoretical results over (PM ) In this section, by considering the graph (PM ) drawn in Figure , we mainly deal with some graph properties, namely diameter, maximum and minimum degrees, girth, degree sequence, irregularity index, domination number, chromatic number and clique number of (PM ). We first recall that for any simple graph , the distance (length of the shortest path) between two vertices u, v of is denoted by d (u, v). If no such path exists, we set d(x, y) := ∞. Actually, the diameter of is defined by . diam() = sup d (x, y) : x and y are vertices of . We thus get the following result. Theorem  The diameter of the graph (PM ) is . Proof By Figure , it is clearly seen that the vertex xk (if l =  then it exists in the graph) of (PM ) is pendant and so the diameter can be figured out by considering the distance d(PM ) (xk , y), where y is one of the other vertices. If l =  in the presentation (), then the vertex x is pendant of the graph (PM ). By Figure , we also see that the vertex x is connected with all the vertices except the vertices of the form yi yi+ ( ≤ i ≤ n – ) in the vertex set. Thus we can reach these vertices by only one edge from the vertices yi ( ≤ i ≤ n). So, we get diam((PM )) = . The degree deg (v) of a vertex v of is the number of vertices adjacent to v. Among all degrees, the maximum degree () (or the minimum degree δ()) of is the number of the largest (or the smallest) degrees in (see []). In our graph, maximum and minimum degrees are obtained as follows. Theorem  The maximum and minimum degrees of the graphs (PM ) are ⎧.

(9) ⎨n + ; if l =  in (), (PM ) = ⎩n; if l =  in ().

(10). and δ (PM ) =. ⎧ ⎨;. if l =  in (),. ⎩; if l =  in (),. respectively. Proof By Figure , it is seen that the vertex x is connected with all vertices of the form yi α α α ( ≤ i ≤ n), y i y i · · · ynin ( ≤ i ≤ n) and the vertex xk if l =  in the presentation (). So, ((PM )) = n + n +  = n + . In the case l =  in the presentation (), since there is no vertex labeled by xk , we get ((PM )) = n + n = n. On the other hand, if l =  in the presentation (), then there is only one edge from the vertex xk to the vertex x. Thus δ((PM )) = . Otherwise, since the vertices which are of the form yi yi+ ( ≤ i ≤ n – ) are connected to the vertices yi and yi+ , we get δ((PM )) = , as required. It is known that the girth of a simple graph is the length of the shortest cycle contained in . However, if the graph does not contain any cycle, then the girth of it is assumed to be infinity. Hence the other result of this section is the following..

(11) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Page 5 of 8. Theorem  The girth of the graph (PM ) is . Proof By the edge definition of (PM ), the vertex x is connected to the vertices of the form α α α yi ( ≤ i ≤ n) and y i y i · · · ynin ( ≤ i ≤ n). There also exists an edge between vertices yi α α α and y i y i · · · ynin . So, the length of the shortest cycle contained in the graph (PM ) is . There also exists the term degree sequence DS(), which is a sequence of degrees of vertices of the graph . Recently, in [], a new parameter for graphs, namely the irregularity index of , has been defined and denoted by MWB(). In fact MWB() is the number of distinct terms in the set DS(). Theorem  The degree sequence and irregularity index of (PM ) are given by.

(12). DS (PM ) =. ⎧ ⎪ (, , , . . . , , , , , , . . . , , n); if l = , ⎪ ⎪ ⎨ n– times. n– times. ⎪ ⎪ (, , . . . , , , , , , . . . , , n); ⎪ ⎩ n– times. if l = . n– times. and ⎧ ⎪ ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪;.

(13) ⎨ MWB (PM ) = ; ⎪ ⎪ ⎪ ⎪ ⎪; ⎪ ⎪ ⎪ ⎩ ;. if l =  and n = , if l =  and n ≥ , if l =  and n = , if l =  and n = , if l =  and n ≥ ,. respectively. Proof Let us consider the case l =  in the presentation (). In this case, since the vertex xk is connected with only the vertex x, then we clearly obtain that the degree of xk is . Now we αj αj αjn consider the vertices of the form yi yi+ ( ≤ i ≤ n – ) and y y · · · yn ( ≤ j ≤ n). Since these vertices are connected with the vertices yi , yi+ ( ≤ i ≤ n – ) and yj ( ≤ j ≤ n), x, the degree of them is . Thus we have n –  vertices which have degree . By Figure , we α see that the vertices y and yn are connected to y y , y , yα  yα  · · · ynn , x and yn– yn , yn– , α α y n y n · · · yαnnn , x, respectively. So, the degree of y and yn is . The remaining vertices yi α α α ( ≤ i ≤ n – ) are connected to the vertices yi , yi– yi , yi yi+ , yi+ , y i y i · · · ynin and x. So, α α α the degree of them is . Since the reminded vertex x is connected with yi , y i y i · · · ynin k ( ≤ i ≤ n) and x the degree of it is n + . Now, let us consider the case l =  in the presentation (). In this case, since the vertex xk does not exist, we get the degree sequence DS((PM )) as depicted in the theorem. For MWB((PM )), we need to consider the number n in the presentation (). The number of distinct terms in the set DS((PM )) depends on the number n. By considering the number n, we can easily obtain the irregularity index MWB((PM )), as required. A subset D of the vertex set V () of a graph is called a dominating set if every vertex V ()\D is joined to at least one vertex of D by an edge. Additionally, the domination.

(14) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. number γ () is the number of vertices in the smallest dominating set for the graph (see []). Theorem  The domination number of (PM ) is given by.

(15) γ (PM ) =. ⎧ ⎨ n+ ; . n is even,. ⎩ n+ ; . n is odd.. Proof By considering Figure , the vertex x is adjacent to all the vertices except the vertex of the form yi yi+ ( ≤ i ≤ n – ). Thus, in the domination set, there must be some vertices of the form yi adjacent to the vertices of the form yi yi+ . These vertices depend on the number of n in the presentation (). If n is even, then there are n + vertices (y , y , . . . , yn , x) +  vertices. Hence the in the smallest dominating set for (PM ). Otherwise, there are n–  result. Basically, the coloring of a graph is to be an assignment of colors (elements of some set) to the vertices of , one color to each vertex, so that adjacent vertices are assigned distinct colors. If n colors are used, then the coloring is referred to as n-coloring. If there exists an n-coloring of , then is called n-colorable. The minimum number n for which is ncolorable is called the chromatic number of and is denoted by χ(). There exists another graph parameter, namely the clique of a graph . In fact, depending on the vertices, each of the maximal complete subgraphs of is called a clique. Moreover, the largest number of vertices in any clique of is called the clique number and is denoted by ω(). In general, it is well known that χ() ≥ ω() for any graph (for instance []). Theorem  The chromatic number χ((PM )) is equal to . Proof Let us consider the vertex x in the graph (PM ) drawn in Figure . It is easy to see that x is adjacent to all the other vertices except the vertices of the form yi yi+ ( ≤ i ≤ n–). That means the color used for the vertex x can be used for the vertices yi yi+ . Thus let us suppose that the color for x and yi yi+ is labeled by C . Next, let us consider the vertices yi ( ≤ i ≤ n) in Figure . Since these vertices are connected with each other doubly, we have two different colors labeled by C and C . In other words, if we label the vertex y by C , then we label the vertex y by C , y by C and so on. Now we take account of vertices I, II, . . . , N . Since these vertices are adjacent to the vertices y , y , . . . , yn , respectively, they can be labeled by C and C , respectively. Since the remaining vertex xk is just connected with the vertex x, it can be labeled by C . Hence the result. We note that the chromatic number of (PM ) does not depend on the number of generators of the free abelian monoid of rank n. Theorem  The clique number ω((PM )) is equal to . Proof By Figure , we have three types of maximal complete subgraphs of (PM ). These types consist of the following edges which have three vertices: yi – yi+ – yi yi+ – yi , yi – α α α yi+ – x – yi and yi – y i y i · · · ynin – x – yi . Hence the result.. Page 6 of 8.

(16) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Page 7 of 8. Figure 2 The graph defined in Example 1.. Example  Let us consider a free abelian monoid of rank , F , and a finite cyclic monoid C with the presentations. PF = y , y , y ; y y = y y , y y = y y , y y = y y and PC = x; x = x , respectively. By taking the matrix ⎡.  ⎢ M = ⎣ .   . ⎤  ⎥ ⎦ . and the homomorphism θ : C → End(F ), we can get a semi-direct product F θ C with the presentation PF θ C = y , y , y , x; y y = y y , y y = y y , y y = y y , x = x , y x = xy y y , y x = xy y y , y x = xy y y . By considering the presentation in () and Figure , we have the following: • V ((PF θ C )) = {y , y , y , x, x , y y , y y , y y y , y y y , y y y } and so |V ((PF θ C ))| = . +  = . • E((PF θ C )) = {ei ( ≤ i ≤ ), in Figure } and so |E((PF θ C ))| = . –  = . • diam((PF θ C )) = . • ((PF θ C )) = . +  =  and δ((PF θ C )) = . • girth((PF θ C )) = . • DS((PF θ C )) = (, , , , , , , , ) and so MWB((PF θ C )) = . = . • γ ((PF θ C )) = +  • χ((PF θ C )) = ω((PF θ C )) = .. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors completed the paper together. All authors read and approved the final manuscript.. ().

(17) Karpuz et al. Journal of Inequalities and Applications 2013, 2013:118 http://www.journalofinequalitiesandapplications.com/content/2013/1/118. Author details 1 Department of Mathematics, Kamil Özdag Science Faculty, Karamanoglu Mehmetbey University, Yunus Emre Campus, Karaman, 70100, Turkey. 2 Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea. 3 Department of Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey. 4 Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey. Acknowledgements Dedicated to Professor Hari M Srivastava. The third and fourth authors are partially supported by Research Project Offices (BAP) of Selcuk (with Project Number 13701071) and Uludag (with Project Numbers 2012-12 and 2012-19) Universities, respectively. The second author is supported by the Faculty Research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea. Received: 28 January 2013 Accepted: 1 March 2013 Published: 20 March 2013 References 1. Cevik, AS, Das, KC, Cangul, IN, Maden, AD: Minimality over free monoid presentations. Hacet. J. Math. Stat. (accepted) 2. Kelarev, AV, Praeger, CE: On transitive Cayley graphs of groups and semigroups. Eur. J. Comb. 24, 59-72 (2003) 3. Kelarev, AV: On Cayley graphs of inverse semigroups. Semigroup Forum 72, 411-418 (2006) 4. Luo, Y, Hao, Y, Clarke, GT: On the Cayley graphs of completely simple semigroups. Semigroup Forum 82, 288-295 (2011) 5. Anderson, DF, Livingston, PS: The zero-divisor graph of commutative ring. J. Algebra 217, 434-447 (1999) 6. Anderson, DF, Badawi, A: The zero-divisor graph of a ring. Commun. Algebra 36(8), 3073-3092 (2008) 7. DeMeyer, FR, DeMeyer, L: Zero-divisor graphs of semigroups. J. Algebra 283, 190-198 (2005) 8. Dlab, V, Neumann, BH: Semigroups with few endomorphisms. J. Aust. Math. Soc. A 10, 162-168 (1969) 9. Cevik, AS: The p-Cockcroft property of the semi-direct products of monoids. Int. J. Algebra Comput. 13(1), 1-16 (2003) 10. Wang, J: Finite derivation type for semi-direct products of monoids. Theor. Comput. Sci. 191(1-2), 219-228 (1998) 11. Gross, JL: Handbook of Graph Theory. Chapman & Hall/CRC, London (2004) 12. Mukwembi, S: A note on diameter and the degree sequence of a graph. Appl. Math. Lett. 25, 175-178 (2012). doi:10.1186/1029-242X-2013-118 Cite this article as: Karpuz et al.: A new graph based on the semi-direct product of some monoids. Journal of Inequalities and Applications 2013 2013:118.. Page 8 of 8.

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