Complement of the generalized total graph of fields. (English) Zbl 1468.05248
Summary: Let \(R\) be a commutative ring and \(H\) be a multiplicative prime subset of \(R\). The generalized total graph \(GT_H (R)\) is the undirected simple graph with vertex set \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if \(x+y \in H\). For a field \(F, H = \{ 0 \} \) is the only multiplicative prime subset of \(F\) and the corresponding generalized total graph is denoted by \(GT(F)\). In this paper, we investigate several graph theoretical properties of \(\overline{GT(F)}\), where \(\overline{GT(F)}\) is the complement of the generalized total graph of \(F\). In particular, we characterize all the fields for which \(\overline{GT(F)}\) is unicyclic, split, chordal, claw-free, perfect and pancyclic.
MSC:
| 05C75 | Structural characterization of families of graphs |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13M05 | Structure of finite commutative rings |
References:
| [1] | Anderson, D. F.; Badawi, A., The total graph of a commutative ring, J. Algebra, 320, 7, 2706-2719 (2008) · Zbl 1158.13001 |
| [2] | Anderson, D. F.; Badawi, A., The generalized total graph of a commutative ring, J. Algebra Appl., 12, 5, 1250212 (2013) · Zbl 1272.13003 |
| [3] | Anderson, D. F.; Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra, 217, 443-447 (1999) · Zbl 0941.05062 |
| [4] | Asir, T.; Tamizh Chelvam, T., On the total graph and its complement of a commutative ring, Comm. Algebra, 41, 10, 3820-3835 (2013) · Zbl 1272.05074 |
| [5] | Asir, T.; Tamizh Chelvam, T., On the intersection graph of gamma sets in the total graph II, J. Algebra Appl., 12, 4, 1250199 (2013) · Zbl 1264.05058 |
| [6] | Badawi, A.; Fontana, M., Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, On the total graph of a ring and its related graphs: a survey, 39-54 (2014), New York: Springer, New York · Zbl 1327.13007 |
| [7] | Balakrishnan, R.; Ranganathan, K., A Text Book of Graph Theory (2000), New York: Springer, New York · Zbl 0938.05001 |
| [8] | Chartrand, G.; Zhang, P., Introduction to Graph Theory (2006), India: Tata McGraw-Hill, India |
| [9] | Földers, S.; Hammer, P. L.; Koffman, F., Proc. 8th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Split graphs, 311-315 (1977), Baton Rouge, LA: Louisiana State Univ, Baton Rouge, LA · Zbl 0407.05071 |
| [10] | Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs (2004), Amsterdam: Elsevier B.V, Amsterdam · Zbl 1050.05002 |
| [11] | Harary, F., Graph Theory (1969), Reading, MA: Addison-Wesley, Reading, MA · Zbl 0182.57702 |
| [12] | Khashyarmanesh, K.; Khorsandi, M. R., A generalization of the unit and unitary Cayley graphs of a commutative ring, Acta Math. Hung., 137, 4, 242-253 (2012) · Zbl 1289.05205 |
| [13] | Kaplansky, I., Commutative Rings (1974), Washington, NJ: Polygonal Publishing House, Washington, NJ · Zbl 0296.13001 |
| [14] | Nazzal, K., Total graphs associated to a commutative ring, Palestine. J. Math, 5, 1, 108-126 (2016) · Zbl 1346.13005 |
| [15] | Tamizh Chelvam, T.; Asir, T., A note on total graph of, J. Discrete Math. Sci. Cryptogr., 14, 1, 1-7 (2011) · Zbl 1222.05094 |
| [16] | Tamizh Chelvam, T.; Asir, T., Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory, 32, 2, 341-354 (2012) · Zbl 1255.05110 |
| [17] | Tamizh Chelvam, T.; Asir, T., On the genus of the total graph of a commutative ring, Commun. Algebra, 41, 1, 142-153 (2013) · Zbl 1261.05042 |
| [18] | Tamizh Chelvam, T.; Asir, T., On the intersection graph of gamma sets in the total graph, J. Algebra Appl., 12, 4, 1250198 (2013) · Zbl 1264.05061 |
| [19] | Tamizh Chelvam, T.; Asir, T., Domination in the total graph of a commutative ring, J. Combin. Math. Combin. Comput., 87, 147-158 (2013) · Zbl 1297.05114 |
| [20] | Tamizh Chelvam, T.; Balamurugan, M., On the generalized total graph of fields and its complement, Palestine J. Math., 7, 2, 450-457 (2018) · Zbl 1393.05224 |
| [21] | Tamizh Chelvam, T.; Balamurugan, M., Complement of the generalized total graph of commutative rings, J. Anal., 27, 539-553 · Zbl 1411.05118 |
| [22] | Tamizh Chelvam, T.; Balamurugan, M., Complement of the generalized total graph of, FILOMAT, 33, 18, 6103-6113 (2019) · Zbl 1499.05296 |
| [23] | West, D. B., Introduction to Graph Theory (2007) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
