Z Halkası Üzerinde Nilpotent Grafların Laplasyan Spektral Özellikleri Sezer Sorgun *1, H. Pınar Cantekin 2

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1 Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Vol(No): pp, year SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ DERGİSİ SAKARYA UNIVERSITY JOURNAL OF SCIENCE e-issn: X Dergi sayfası: Geliş/Received Kabul/Accepted Doi Z Halkası Üzerinde Nilpotent Grafların Laplasyan Spektral Özellikleri Sezer Sorgun *1, H. Pınar Cantekin 2 ÖZ birimli bir halka olsun. ' nin ile gösterilen nilpotent grafı, = { : ç } noktalar kümesi ve, halkasının bütün nilpotent elemanlarının kümesi olmak üzere; iki farklı ve noktaları komşudur önermesini sağlayan kenarlar kümesinden oluşur. Bu makalede Z halkası üzerinde tanımlanan nilpotent grafın Laplasyan spektral özellikleri incelenmektedir. Anahtar Kelimeler: Nilpotent graf, Laplasyan matris, Spektrum ABSTRACT We consider a ring with unity. The nilpotent graph of is a graph with vertex set = { : }; and two distinct vertices and are adjacent iff, where is the set of all nilpotent elements of and it is denoted by. In this paper we study Laplacian spectral properties of the nilpotent graph over the ring Z. Keywords: Nilpotent Graph, Laplacian matrix, Spectrum * Sorumlu Yazar / Corresponding Author 1 Nevşehir HBV Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü NEVŞEHİR-srgnrzs@gmail.com 2 Nevşehir HBV Üniversitesi, Fen Edebiyat Fakültesi Matematik Bölümü NEVŞEHİR- pinarcantekin007@gmail.com 2017 Sakarya Üniversitesi Fen Bilimleri Enstitüsü

2 1. INTRODUCTION Let = (, ) be a graph with vertex set = {,,,, } and edge set. The degree of a vertex is the number of vertices which adjacent to and denoted by. The set of neighbors of is denoted by. Let max = and min =. When = (, ) and = (, ) are given graphs on disjoint sets of vertices, their union is the graph + = (, ). The join,, is the graph obtained from + by adding new edges from each vertex of to every vertex of. A complete split graph is a graph on vertices consisting of a clique on vertices and a stable set on the remaining vertices in which each vertex of the clique is adjacent to each vertex of the independent set and denoted by (, ). The Laplacian matrix of a graph is denoted by =, where = (,,, ) and are the diagonal matrix of vertex degrees and (0,1) adjacency matrix of, respectively. It is well known that is positive semidefinite symmetric and singular. Its eigenvalues of are denoted by, 1 and these eigenvalues can be ordered by = 0. The Laplacian spectrum of the graph and the Laplacian characteristic polynomial are defined by = {,,, } and Φ, respectively. Recently, one of the interesting research topic has become the combination some algebraic structures and graph. There can be found a lot of papers on connecting a graph to a ring [1-7]. We consider a ring with unity. The nilpotent graph of a ring is introduced in [5]. That is; the nilpotent graph of a ring is a graph with vertex set = {0 : forsome 0 } such that two distinct vertices and are adjacent iff and denoted by Γ. When given the ring = Z, it is well known that it has a nonzero nilpotent element if and only if is divisible by the square of some primes. From this fact, Z does not have any non-zero nilpotent element when is prime or =. Also, it is easily seen that (Z ) = {0,, 2,, ( 1) } (1) when =, > 1 and ( ), 2( ), (Z ) = (2), ( 1)( ) when =, 2. In literature, although there are a few studies related to graph parameters such as diameter, girth, etc. on the nilpotent graph, there is no study on the (Laplacian) spectral properties of the nilpotent graphs. In this paper we use the nilpotent graph over the Z ring for all and examine Laplacian spectral properties of it. We have seen that almost all Laplacian eigenvalues of the graph are exactly the degrees of the vertex (of course, integers). 2. MAIN RESULTS Lemma 2.1. Let Z be integer ring, where = and is a prime number. Then, the vertex set of Γ (Z ) is (Z ) = Z (3) Moreover, we have = 2 for (Z ) and = 1 for (Z ). We have (Z ) = 0,, 2, 3,, ( 1), where is a prime number such that. Let Z. Then, (Z ) since is nilpotent element of Z for all (Z ), we get (Z ). Hence Z (Z ). It is clear that (Z ) Z from the definition of the vertex set of a nilpotent graph. Hence, we get (Z ) = Z. It is easy to see = 2 for (Z ) by the same arguments above. Now let us show that = 1 for (Z ). Let (Z ) and any Z. There are two cases. Case 1: (Z ). In this case, we say that is nilpotent ( ) for all Z. Then, we have = { : (Z )} = 1. Case 2: Z. Now we consider that (Z ). Then, =, where 1 1. Hence Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol., no. : pp, year 2

3 or. If, then (Z ). But this is a contradiction that (Z ). Similarly, if, then we get the same contradiction to (Z ). Therefore, we have (Z ), that's;. Consequently, we get = { : (Z )} = 1. (4) for all (Z ). Remark 2.2. By Lemma 2.1, we see that Γ (Z ) has two distinct degrees such that Δ = 2 and = 1. Theorem 2.3. If is a prime number then Γ Z = (0,, (Δ + 1) ) (5) where Δ = 2 and = 1. Let =,,,,,, be eigenvector corresponding to any eigenvalue for L(Γ (Z )). We have = (6) = : (7) for = 1,,,, 1. Now we consider vectors = such that (Z ) and = 1 for all other Z and hence we get ( 1)-vectors as ( 1,, 1,, 1,, 1), ( 1,, 1,, ( 1,, 1,, 1,, 1),..., 1,, 1). These vectors are the characteristic eigenvectors corresponding eigenvalue 1 of Γ Z and each provide the equality in (7). Since all vectors above are linear independent, the algebraic multiplicity of the eigenvalue 1 is ( 1). By Remark 2.2, we get that Δ + 1 is the eigenvalue with 1 multiplicity. Similarly, we consider vectors as = 1 such that (Z ) and = 1 for only one,. We get, 1 = 1 Therefore we choose ( 1) linear independent vectors as 1,0,,0, 1, 0,,0,, 0,,0, 1, 0,,0, 1, 0,,0 (8) Each vectors provide the equality in (7) and they are also the characteristic eigenvectors corresponding eigenvalue ( 1) of Γ Z. By again Remark 2.2, we get that Δ + 1 is the eigenvalue with Δ + 1 multiplicity. On the other hand, it is well known that vector = (1,1,,1) is an eigenvector corresponding eigenvalue 0. Hence we get the result of the theorem. Lemma 2.4. [8] Let be a connected graph on vertices, then has three distinct eigenvalues 0,,,,,, and, with multiplicities and if and only if i. ( 1), ii. + 1, iii. =, where is isolated vertices with multiplicity. Remark 2.5. The graph Z has three distinct eigenvalues. Hence, by Lemma 2.4., we get Z (, ). Lemma 2.6. Let Γ (Z )be graph, where =, 2. i. If = 1 for each, then (Z ) = (9) where = { :1 1} and = {1,2,, }. ii. If 2 for at least, then (Z ) = Z (10) i. Let = 1 for each. In this case, there is no non zero nilpotent element of Z. That's; (Z ) = {0}. Let 0 (Z ). From the definition of the vertices set for nilpotent graphs, there is a non-zero element Z such that = 0. (11) By (11), we get and hence (Z ). It is also clear that (Z ). From these inclusions, we have the required result. ii. If 2 for at least, then we have (Z ) = {( ), 2( ),, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol., no. : pp, year 3

4 ( 1)( )} by (2). Then, it is easily shown that each non-zero class of Z is adjacent to any nilpotent element. This completes the result. Remark 2.7. From Lemma 2.6. (i)., we can easily see that the number of vertices of Γ (Z ) is 1, where is the number of positive divisors of. Particularly, Γ (Z ), for, primes and Γ (Z ), where is a null graph. Lemma 2.8. Let's consider the graph Γ (Z ), where =, 2. i. = 2 for all (Z ) ii. If for (Z ) such that,, for 1 and = {1,2,, }, then we get =, 2,, 1 (12) = 1 (13) where = for every {,, }; and are the degree of vertex and the set of neighbors of, respectively. iii. If (, ) = 1 for all 1, then we get = (Z ) = {( ), 2( ),, ( 1)(( )} (14) = 1 (15) i. By similar method in Lemma 2.1., for all (Z ), is also nilpotent element of Z such that (Z ). Then; ~ for every. That is; = { : Z } (16) = 2 (17) ii. It is clear that ~ under the mentioned conditions. iii. If (, ) = 1, ~ for any Z if and only if (Z ). From this fact, we get the result directly. Theorem 2.9. Let Z be a ring, where =. Then we have ( Φ ( (Z )) = ) ( + 2) where = ( ), =, ( ),..., =, ( ) and is any polynomial such that is the cardinality of the set { Z : = }; (, ) = 1 for 1 and is Euler's totient function. We consider at least 2. Let be eigenvector corresponding to for (Γ (Z )). We have = = : (19) for (Z ). There are three cases for vertex from Lemma 2.8. Let (Z ). We consider = 1 and = 1 for one (Z ). Hence these ( 1)-vectors are the characteristic vectors corresponding to eigenvalue = 2 (by Lemma 2.8.(i).) and provide equality in (19). Let (, ) = 1, 1. Then we get = 1 from Lemma 2.7.iii.. Similarly, if we consider vectors = 1 and = 1 for one (Z ), such that = and (, ) = 1. Since =, hence there are -class which are relative prime. The other arguments can be shown by the same methods. APPENDICES Now we give Laplacian spectrum of the nilpotent graph over ring Z, for As seen below table, almost all Laplacian eigenvalues of the graphs are the integers. n Laplacian Spectrum of Γ (Z ) 4 {0, 1, 3} 8 {0, 3, 7 } 9 {0, 2, 8 } 12 {0, 1, 5, 7, 11 } 16 {0, 7, 15 } 18 {0, 2, 5, 8, 11, 17 } 20 {0, 1, 3, 9, 11, 19 } 24 {0, 3, 7, 11, 15, 23 } 25 {0, 4, 24 } 27 {0, 8, 26 } 28 {0, 1, 3, 13, 15, 27 } 32 {0, 15, 31 } 36 {0, 5, 11, 17, 15, 23, 35 } 40 {0, 3, 7, 19, 23, 39 } 44 {0, 1, 3, 21, 23, 43 } 45 {0, 2, 8, 14, 20, 44 } 48 {0, 7, 15, 23, 31, 47 } 49 {0, 6, 48 } Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol., no. : pp, year 4

5 n Laplacian Spectrum of Γ (Z ) 50 {0, 4, 9, 24, 29, 49 } 54 {0, 8, 17, 26, 35, 53 } 56 {0, 3, 7, 27, 31, 55 } {0,1, , 3, 4.4, 5, 9, 11, 13.3, 19, 23.6, 29, 31.26, 59 } 63 {0, 2, 8, 20, 26, 62 } 72 {0, 11, 23, 35, 47, 71 } 75 {0, 4, 14, 24, 34, 74 } 80 {0, 7, 15, 39, 47, 78 } 81 {0, 25, 80 } 88 {0, 3, 7, 43, 47, 87 } {0, 2 90, 3.94, 5, 7.16, 8, 14, 17, 20.45, 29, 36.04, 44, 47.39, 89 } 98 {0, 6, 13, 48, 55, 97 } 99 {0, 2, 8, 32, 38, 98 } 100 {0, 9, 19, 49, 59, 99 } REFERENCES [1] D.F. Anderson, P.S. Livingston, "The zerodivisor graph of a commutative ring", J. Algebra, vol. 217, pp , [2] D.F. Anderson, A. Badawi, "The total graph of a commutative ring", J. Algebra, vol. 320, pp , [3] M.J.Nikmehr, S.Khojasteh, "On the nilpotent graph of a ring", Turkish Journal of Mathematics, vol. 37, pp , [4] P. Sharma, A.Sharma, R.K. Vats, "Analysis of Adjacency Matrix and Neighborhood Associated with Zero Divisor Graph of Finite Commutative Rings", International Journalof Computer Applications, vol. 13 pp , [5] P.W. Chen, A kind of graph structure of rings, Algebra Colloq., vol. 10 (2),pp , [6] S. Akbari, A. Mohammadian, "Zerodivisor graphs of non-commutative rings", J. Algebra, vol. 296, pp , [7] S.Akbari, A. Mohammadian, "On zerodivisor graphs of finite rings", J. Algebra, vol. 13, pp , [8] S. J. Kirkland, "Completion of Laplacian integral graphs via edge additions", Discrete Math, vol. 295, pp , Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol., no. : pp, year 5