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On bipartite divisor graph for character degrees | ||
| International Journal of Group Theory | ||
| مقاله 37، دوره 6، شماره 1، خرداد 2017، صفحه 1-7 اصل مقاله (188.24 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.9852 | ||
| نویسنده | ||
| Seyed Ali Moosavi* | ||
| University of Qom | ||
| چکیده | ||
| The concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95--105.]. In this paper, we will consider this graph for the set of character degrees of a finite group $G$ and obtain some properties of this graph. We show that if $G$ is a solvable group, then the number of connected components of this graph is at most $2$ and if $G$ is a non-solvable group, then it has at most $3$ connected components. We also show that the diameter of a connected bipartite divisor graph is bounded by $7$ and obtain some properties of groups whose graphs are complete bipartite graphs. | ||
| کلیدواژهها | ||
| bipartite divisor graph؛ character degree؛ connected component؛ diameter | ||
| مراجع | ||
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[2] A. Heydari and B. Taeri, Onnite groups with two irreducible character degrees, Bul l. Iranian Math. Soc., 34 (2008) 39-47.
[3] D. Bubb oloni, S. Dol, M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, J. Pure Appl. Algebra, 213 (2009) 1722-1734.
[4] M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95-105.
[5] M. A. Iranmanesh, Bipartite divisor graph: A survey, Electron. Notes Discrete Math., 45 (2014) 43-49.
[6] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, San Diego, California, 1976.
[7] M. L. Lewis, An overview of graphs asso ciated with character degrees and conjugacy class sizes innite groups, Rocky Mountain J. Math., 38 (2008) 175-211.
[8] M. L. Lewis, Solvable groups whose degree graphs have two connected comp onents, J. Group Theory., 4 (2001) 255-275.
[9] M. L. Lewis, A solvable group whose character degree graph has diameter 3, Proc. Amer. Math. Soc., 130 (2002) 625-630. | ||
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