Toroidality and Projectivity index of Jacobson Graph
Subject Areas : StatisticsAtossa Parsapour 1 , Khadijeh Ahmad Javaheri 2
1 - Department of MathematicsBandar Abbas BranchIslamic Azad UniversityBandar AbbasIran
2 - Department of Mathematics,Bandar Abbas Branch.Islamic Azad University.Bandar Abbas.Iran
Keywords: گراف ژاکوبسون, گراف خط مکرر, اندیس چنبره ای, اندیس تصویری,
Abstract :
Given a graph Г, we denote the kth iterated line graph of Г by LK(Г) and LK(Г)=L(LK-1(Г)). In particular, L0(Г)=Г and L1(Г)=L(Г) is the line graph of Г. The toroidality (and projectivity) index of a graph Г is the smallest integer k such that the kth iterated line graph of Г is non-toroidal (and non-projective). We denote the toroidality index of a graph Г by ξT and the projectivity index of a graph Г by ξP. If LK(Г) is toroidal (and projective) for all k≥0, we define ξT=∞ (and ξP=∞). Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a simple graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. In this paper, we study the toroidality and projectivity indices of Jacobson graphs. We give full characterization of this graph with respect to its toroidality and projectivity indices. Moreover, the toroidality and projectivity index of Jacobson graph is either infinite or two.
[1] D. Archdeacon. A Kuratowski theorem for the projective plane, J. Graph Theory 5:243-246(1981)
[2] M.F. Atiyah and I.G. MacDonald. Introduction to Commutative Algebra, Addision-Wesley, Reading, Mass. (1969)
[3] A. Azimi, A. Erfanian and M. Farrokhi D.G.. The Jacobson graph of commutative rings, J. Algebra Appl. 12: 1250179-1250197(2013)
[4] R. Bodendiek and K. Wagner. On the minimal basis of the spindle-surface, Combinatorics and Graph Theory, Banach Center Publications, 25 PWN-Polish Scientific Publishers, Warsaw (1989)
[5] A. Bouchet. Orientable and nonorientable genus of the complete bipartite graph, J. Combin. Theory Ser. B 24:24-33(1978)
[6] H.J. Chiang-Hsieh, P.F. Lee and H.J. ang. The embedding of line graphs associated to the zero-divisor graphs of commutative rings, Israel J. Math. 180:193-222(2010)
[7] B. Corbas and G.D. Williams. Rings of order p5. Part I. Nonlocal rings, J. Algebra 231:677-690(2000)
[8] R.L. Hemminger and L.W. Beineke. Line graphs and line digraphs, L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory I, Academic Press, London (1978)
[9] J. Fiedler, J.P. Huneke, R.B. Richter and N. Robertson. Computing the orientable genus of projective graphs, J. Graph Theory 20:297-308(1995)
[10] H. Glover, J.P. Huneke and C.S. Wang. 103 graphs that are irreducible for the projective plane, J. Combin Theory Ser. B 27:332-370(1979)
[11] J. L. Gross and T.W. Tucker. Topological graph theory, John Wiley and Sons Inc., New York (1987)
[12] I. Kaplansky. Commutative Rings, Revised Edition, University of Chicago Press, Chicago (1974)
[13] K. Kuratowski, Sur le probléme des courbes gauches en topologie, Fund. Math. 15:271-283(1930)
[14] W.S. Massey. Algebraic Topology, An Introduction, Harcourt-Brace, Orlando, FL (1967)
[15] W. Myrvold and J. Roth. Simpler projective plane embedding, Ars Combin. 75:135-155(2005)
[16] A. Parsapour, K. Khashyarmanesh, M. Afkhami and Kh. Ahmad Javaheri. Classification of finite commutative rings with planar, toroidal and projective line graphs associated to jacobson graphs, Math. Notes 98:813-819(2016)
[17] G. Ringel. Map Color Theorem, Springer-Verlag, New York/Heidelberg (1974)
[18] A.T. White. Graphs, groups and surfaces, North-Holland Publishing Co., Amster-dam, second ed. (1984)
[19] H. Whitney. Congruent graphs and connectivity of graphs, Amer. J. Math. 54: 150-168(1932)
[20] D. Zeps. Förbidden minors for projective plane are free-toroidal or non-toroidal, IUUK-CE-ITI series (2009)
