The independence polynomial of conjugate graph and noncommuting graph of groups of small order.
DOI:
https://doi.org/10.11113/mjfas.v0n0.709Keywords:
Independence polynomial, conjugate graph, noncommuting graph, nonabelian groupAbstract
An independent set of a graph is a set of pairwise non-adjacent vertices. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate. The noncommuting graph is defined as a graph whose vertex set is non-central elements in which two vertices are adjacent if and only if they do not commute. In this research, the independence polynomial of the conjugate graph and noncommuting graph are found for three nonabelian groups of order at most eight.
References
Abdollahi, A., S. Akbari, and H. R. Maimani. 2006. Non-Commuting Graph of a Group. Journal of Algebra. 298(2006): 468-492.
Asghar Talebih, A. 2008. On the Non-commuting Graphs of Group International Journal of Algebra. 2(20): 957-961.
Balakrishnan, R. and K. Ranganathan. 2012. A Textbook of Graph Theory. 2nd ed. New York: Springer.
Bertram, E. A. 1983. Some Applications of Graph Theory to Finite Groups. Discrete Mathematics. 44: 31-43.
Darafsheh, M. R. 2009. Groups with the Same Non-commuting Graph. Discrete Applied Mathematics. 157: 833-837.
Erfanian, A. and B. Tolue. 2012. Conjugate Graphs of Finite Groups. Discrete Mathematics, Algorithms and Applications. 4(2): 35-43.
Erfanian, A., F. Mansoori and B. Tolue. 2015. Generalized Conjugate Graph. Georgian Mathematical Journal. 22(1): 1-8.
Ferrin, G. 2014. Independence Polynomials. Master Dissertation. University of South Carolina.
Fraleigh, J. B. 2003. A First Course in Abstract Algebra. 7th ed. U.S.A.: Pearson Education, Inc.
Hoede, C. and X. Li. 1994. Clique Polynomials and Independent Set Polynomials of Graphs. Discrete Mathematics. 125: 219-228.
Levit, V. E. and E. Mandrescu. 2005. The Independence Polynomial of a Graph – A Survey. Proceedings of the 1st International Conference on Algebraic Informatics. 233254.
Moghaddamfar, A. R., W. J. Shi, W. Zhou and A. R. Zokayi. 2005. On the Noncommuting Graph Associated with a Finite Group. Sherian Mathematical Journal. 46(2): 325-332.
Moradipour K., N. H. Sarmin and A. Erfanian. 2013. On Non-commuting Graphs of Some Finite Groups. International Jrnal of Applied Mathematics and Statistics. 45(15): 473-476.
Omer, S. M. S., N. H. Sarmin and A. Erfanian. 2013. The Probability that an Element of a Group Fixes a Set and Its Graph Related to Conjugacy Classes. Journal of Basic Applied Scientific Research. 3(10): 369-380.
Omer, S. M. S., N. H. Sarmin and A. Erfanian. 2013. The Probability that an Element of a Symmetric Group Fixes a Set and Its Application in Graph Theory. World Applied Sciences Journal. 27(12): 1637-1642.
Rose, H. E. 2009. A Course on Finite Groups. London: Springer-Verlag.
Rosen, K. H. 2013. Discrete Mathematics and Its Applications. 7th ed. New York: McGraw-Hill.
Rotman, J. J. 2003. Advanced Modern Algebra. USA: Prentice Hall.
Sarmin, N. H., I. Gambo and S. M. S. Omer. 2015. The Conjugacy Classes of Metabelian Groups of Order at Most 24. Jurnal Teknologi. 77(1) : 139-143.
Sarmin, N. H., M. Jahandideh and M. R. Darafsheh. 2016. Recent Advances in Commutativity Degrees and Graphs of Groups. Chapter 1: 1-8. Penerbit UTM Press.
Sarmin, N. H., N. H. Bilhikmah, S. M. S. Omer and A. H. M. Noor. 2016. The Conjugacy Classes, Conjugate graph and Conjugacy Class Graph of Some Finitie Metacyclic 2-groups. Menemui Matematik. 38(1): 1-12.
