SOME ASPECTS OF PRIME GRAPH OF SOME RINGS

Transcription

1 SOME ASPECTS OF PRIME GRAPH OF SOME RINGS An Abstract Submitted to Gauhati University for the Degree of Doctor of Philosophy in Mathematics in the Faculty of Science Submitted by Sanjoy Kalita 2014

2 SYNOPSIS Graph theory is a very popular and rapidly growing area of discrete mathematics because of its numerous theoretical development and countless applications to practical problems. As a research area, graph theory is still relatively young, but it is maturing rapidly with many deep results having been discovered over the last couple of decades. The meaning of the term graph in graph theory is not same as the term graph that used to represent a function or statistical data. In graph theory, a graph G is a non-empty finite set of objects called vertices together with a set of unordered pairs of distinct vertices of G called edges. The vertex set of G is denoted by V(G) and the edge set is denoted by E(G). In mathematics, graphs are useful in geometry, algebra and certain parts of topology such as knot theory. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road. Because of its multifarious applications in so many areas of mathematical research, Graph Theory drew the attention of mathematical world and has emerged as one of the new branch of mathematics in present days. The use of algebraic techniques in the study of graphs and vice-versa has also acquired importance in the present era. In the last decade of 19th century the literature of algebraic graph theory has grown enormously and many research papers have appeared in this area of graph theory. Algebraic graph theory is a fascinating subject concerned with the interplay between algebra and graph theory. Algebraic tools can be used to give surprising and elegant proofs of graph theoretic facts, and there are many interesting algebraic objects associated with graphs. In recent years algebraic graph theory has become an interesting topic of research. Many algebraic structures are characterized by translating them into graphs. The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Different matrices associated to a graph are mostly used algebraic structure which helps to characterize the graph. Especially, it involves the study of the spectrum 1

3 SYNOPSIS 2 of the adjacency matrix or the Laplacian matrix of a graph. A graph can be completely determined by the adjacency matrix and the spectral properties of this matrix are closely related to properties of the graph. For example if a graph is regular then the eigenvalues of the adjacency matrix of the graph are bounded in absolute values by the degree of the graph. In the case of line graph, there is a strong lower bound for the eigenvalues. Incidence matrix, Cycle matrix, Diagonal matrix are some other matrices associated to a graph which help to characterize the graph. This branch of algebraic graph theory is categorized as spectral graph theory. Another branch of algebraic graph theory includes the study of symmetry and regularity properties of graphs which can be studied using group theory. A symmetry property of a graph is related to the existence of automorphism. The concept of automorphisms of a graph, which is the permutation of vertices that preserves adjacency, played an important role in the characterization of graphs. The existence of automorphisms led to the idea of permutation group of graph. The set of all automorphisms of a graph forms a permutation group. Many properties of permutation group in group theory also exist for permutation group of graphs. One such property is the transitive property. These theories help us to characterise the graphs. The focus of this branch was on symmetry of graphs and based on their symmetry, different families of graphs such as symmetric graphs, vertex-transitive graphs, edge-transitive graphs, distance-transitive graphs, distance-regular graphs, and strongly regular graphs are defined. Many researchers are interested towards the automorphism group of graphs and their characteristics. In recent times many researchers are showing their interest towards the relation of graphs with semigroup, rings and modules. Different kinds of graphs are defined to study these algebraic structures and many important results are established. In this area some significant contributions are made by the researchers like I. Beck[12], D. D. Anderson and M. Naseer[3], P. K. Sharma and S. M. Bhatwadekar[39], D. F. Anderson[4 6], P. S. Livingston[5], S. P. Redmond[35 37], A. V. Kelarev[24], S. Akbari[2], H. R. Maimani[30], S. E. Atani[7, 8], A. Y. Darani[7], F. DeMeyer[18, 19], P. W. Chen[16], S. B. Mulay[33, 34], A. Badawi[4], Halaš, Jukl[21], M. Behboodi[13, 14], Rakeei[13, 14], Spiroff [41], M. Axtell[9, 10, 43], Stickles[9, 10, 43], J. Coykendall[10], M. Afkhami[1], K. Khashyarmanesh[1], J. Han[22] B. Satyanarayana[38]. Another kind of graph structure associated to a ring is studied by Satyanarayana et al[38] and the graph is known as Prime Graph. Our main objective in this thesis is to study the Prime Graph of any finite commutative ring and non-commutative ring.

4 SYNOPSIS 3 Chapter 1 is an introductory chapter. The background to this work, preliminary concept and terminologies of graph, the concept of prime graph, concept of ring and ideal are discussed in this chapter. A survey of some important and interesting results in the field of algebraic graph theory associated to rings and ideals are discussed in this chapter. In Chapter 2 chromatic number of the prime graph of different rings are studied. First section deals with the study of the chromatic number of some commutative rings. The chromatic number of the matrix rings M 2 (R) and M 2 (Z n ) and their relation with the chromatic number of the rings R and Z n are studied in Section 2 and Section 3. Some parts of this paper was presented in "National Conference on Frontiers in Mathematics (NCFM- 2011), Gauhati University (under UGC-SAP)" and the chapter is published in "International Journal of Combinatorial Graph Theory and Applications(IJCGTA)" as a research paper. In Chapter 3 we particularly studied the prime graph of the ring Z n for different values of n and ring Z n [x]/ f(x) for different values of n and f(x) = x 2 + bx+c, b 0, c 0. For the ring Z n [x]/ f(x) we have studied the structure of the prime graphs for both reducible and irreducible polynomials f(x) and different values of n particularly n = p and p n for n Z and n > 0. Some parts of this paper was presented in "International Conference on Recent Advances in Mathematical Statistics and Its Applications in Applied Sciences", Gauhati University held on 31st December, nd January, In Chapter 4 we have considered the finite cartesian product of commutative rings R = R 1 R 2 R n and investigated relation between the chromatic numbers of the ring R and each of the commutative rings R i. Some part of this paper was presented in "78th Annual Conference of the Indian Mathematical Society, Banaras Hindu University- 2013" and the chapter is published in International Journal of Computer Application(IJCA)" as a paper. In Chapter 5, the definition of prime graph is modified by defining the prime graph with respect to its ideals for studying some properties of prime graph. To keep parity with the definition of prime graph we call this graph as prime graph of a ring with respect to ideal. We denote this graph as PG I (R). Some parts of this paper was presented in "57th Annual Technical Session, Assam Science Society, Gauhati University, 2012" and the chapter is published in "International Journal of Mathematical Archive(IJMA)" as a paper.

5 SYNOPSIS 4 In Chapter 6 we have modified the definition of prime graph for any ring, given by Satyanarayana, and we have also defined a directed prime graph for a non-commutative ring. In this Chapter we have studied different properties of these graphs. Some parts of this paper were presented in 3rd International Conference on Frontiers of Mathematics and Application, Burdwan University-2014".

6 Bibliography [1] Afkhami, M.; Khashyarmanesh, K.; On the Cozero-Divisor Graphs of Commutative Rings and Their Complements, Bull. Malays. Math. Sci. Soc. Vol. 35(2012), No. 4, [2] Akbari, S.; Mohammadian A.; On The Zero-Divisor Graph of A Commutative Ring, J. Algebra, Vol. 274(2004), [3] Anderson, D. D.; Naseer, M.; Beck s Coloring of A Commutative Ring,, J. Algebra Vol. 159(1993), [4] Anderson, D. F.; Badawi, A; On The Zero-Divisor Graph of A Ring, Communications in Algebra, Vol. 36(2008), [5] Anderson, D. F.; Livingston, P. S; The zero-divisor graph of a commutative ring, J. Algebra Vol. 217(1999), No. 2, [6] Anderson, D. F.; Mulay, S. B.; On The Diameter and Girth of A Zero-divisor Graph, Journal of Pure and Applied Algebra, Vol. 210(2007), [7] Atani, S. E.; Darani, A. Y.; Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings, Int. J. Contemp. Math. Sci., 26(2007), [8] Atani, S. E.; Farzalipour, F.; Zero-Divisor Graphs of Idealizations with Respect to Modules, Chiang Mai J. Sci., Vol. 36(2009), No. 1, 5-8. [9] Axtell, M.; Stickles, J.; Zero-Divisor Graphs Of Idealizations, Journal of Pure and Applied Algebra, Vol. 204(2006), [10] Axtell, M.; Stickles, J.; Coykendall, J.; Zero-divisor graphs of polynomial and power series over commutative rings, Comm. Alg., Vol. 33(2005),

7 SYNOPSIS 6 [11] Azarpanah, F.; Motamedi, M.; The Zero-Divisor Graph of C(X), Acta Math. Hungar, Vol. 108(2005), No. 1-2, [12] Beck, I.; Coloring of commutative rings, J. Algebra, Vol. 116(1988), No. 1, [13] Behboodi, M.; Rakeei, Z.; The Annihilating-Ideal Graph Of Commutative Rings I, J. Algebra Appl., Vol. 10(2011), No. 4, [14] Behboodi, M.; Rakeei, Z.; The Annihilating-Ideal Graph Of Commutative Rings II, J. Algebra Appl., Vol. 10(2011), No. 4, [15] Biggs, N.; Algebraic Graph Theory, Second Edition, Cambridge Mathematical Library, Cambridge University Press, [16] Chen, P.W.; A Kind Of Graph Structure Of Rings, Algebra Colloq., Vol. 10(2003), No. 2, [17] Christofides, N.; Graph Theory - An Algorithmic Approach, Academic Press London New York San Francisco, [18] Demeyer, L.; D sa, M.; Epstein, I.; Geiser, A.; Smith, K.; Semigroups And The Zero Divisor Graph, Bull. Inst. Combin. Appl., Vol. 57(2009), [19] Demeyer, L.; Greve, L.; Sabbaghi, A.; Wang, J.; The Zero-Divisor Graph Associated To A Semigroup, Comm Algebra. Vol 38(2010), [20] Dheena, P.; Elavarasan, B.; A Generalised Ideal Based Zero Divisor Graphs of Near Ring, Commun. Korean Math. Soc., Vol. 24(2009), No. 2, [21] Halas, R.; Jukl, M.; On Beck s Coloring of Posets,, Discrete Math., Vol. 309(2009), [22] Han; J.; The Zero-Divisor Graph Under Group Actions In A Non-commutative Ring, J. Korean Math. Soc., Vol. 45(2008), No. 6, [23] Harary, F.; Graph Theory, Eddison Wesley Publishing Company inc [24] Kelarev, A. V.; Directed Graphs And Nilpotent Rings, J. Austral. Math. Soc.(Series A), Vol. 65(1999), [25] Kelarev, A. V.; Quinn, S. J.; Directed graph and combinatorial properties of semigroups, J. Algebra, Vol. 251(2002),

8 SYNOPSIS 7 [26] Lam, T.Y.; A First Course In Noncommutative Rings, Springer, New York, [27] Lambek, J.; Lectures on Rings and Modules, Blaisdel Publ. Co [28] Li, A. H.; Li, Q. S.; A Kind Of Graph Structure On Non-Reduced Rings, Algebra Colloq., Vol. 17(2010), No. 1, [29] Li, A. H.; Li, Q. S.; A Kind Of Graph Structure On Von-Neumann Regular Rings, International J. Algebra, Vol. 4(2010), [30] Maimani, H. R.; Median and Center of Zero-Divisor Graph of Commutative semigroups, Iranian Journal of Mathematical Sciences and Informatics Vol. 3(2008), No. 2, [31] Maimani, H. R.; Salimi, M.; Sattari, A.; Yassemi, S.; Zero-Divisor Graph with Respect to an Ideal, Communications in Algebra, Vol. 34(2006), No. 3, [32] Maimani, H. R.; Salimi, M.; Sattari, A.; Yassemi, S.; Co-maximal graph of commutative rings, J. Algebra, Vol. 319(2008), No. 4, [33] Mulay, S. B.; Rings Having Zero-Divisor Graphs Of Small Diameter Or Large Girth, Bulletin of the Australian Math. Society, Vol. 72(2005), No. 3, [34] Mulay, S. B.; Cycles and Symmetries of Zero Divisors, Comm. Algebra, Vol. 30(2002), [35] Redmond, S. P.; An Ideal-Based Zero Divisor Graph of a Commutative Ring, Communications In Algebra, Vol. 31(2003), [36] Redmond, S. P.; Central Sets and Radii of the Zero Divisor Graphs of Commutative Rings, Communications In Algebra, Vol. 34(2006), [37] Redmond, S. P.; Cut Vertices and Degree One Vertices of Zero Divisor Graphs, Communications In Algebra, Vol. 40(2012), [38] Satyanarayana, B.; Syam Prasad, K.; Nagaraju, D.; Prime Graph of a Ring, J. of Combinatorics, Information and System Sources, Vol. 35(2010), No 1-2, [39] Sharma, P. K.; Bhatwadekar, S. M.; A Note On Graphical Representation of Rings, J. Algebra, Vol. 176(1995), [40] Singh, S.; Zameeruddin, Q.; Modern Algebra, (Third Reprint), Vikas Publishing House Pvt. Ltd., Delhi, 1995.

9 SYNOPSIS 8 [41] Spiroff, S.; Wickham, C.; A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors, Comm. In Alg., Vol. 39(2011), [42] Wang, H. J.; Graphs associated to co-maximal ideals of commutative rings, J. Algebra, Vol. 320(2008), No. 7, [43] Warfel, J.; Axtell, M.; Stickles, J.; Zero-divisor graphs of direct products of commutative rings, Houston J. of Math., Vol. 32(2006), [44] Wu, T.; On Directed Zero-Divisor Graphs of Finite Rings, Discrete Math., Vol. 296(2005), No. 1, Prof. Kuntala Patra Supervisor, Department of Mathematics, Gauhati University. Candidate s Signature