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Abstract
Let G be a finite group. Associate a simple undirected graph Γ_G with G, called bipartite graph associated to elements and cosets of subgroups of G, as follows : Take G ∪ S_G as the vertices of Γ_G, with S_G is the set of all subgroups of a group G and join two vertices a ∈ G and H ∈ S_G if and only if aH = Ha. In this paper, hamiltonicity and eulerianity of Γ_G for some finite groups G are studied. In particular, it is obtained that for any cyclic group G, Γ_G is hamiltonian if and only if |G| = 2 and Γ_G is eulerian if and only if |G| is even non-perfect square number. Also, we prove that Γ_Dn is eulerian if k is even and n = 2k and for some other cases of n, Γ_Dn is not eulerian.
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References
- Dummit, D. S., Foote, R. M., Abstract Algebra Third Edition, John Wiley and Sons, Inc : United States of America, 2004.
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- Al-Kaseasbeh, S., Erfanian, A., ”A bipartite graph associated to elements and cosets of subgroups of a finite group”, AIMS Mathematics 2021, 6(10), 10395–10404
References
Dummit, D. S., Foote, R. M., Abstract Algebra Third Edition, John Wiley and Sons, Inc : United States of America, 2004.
Jensen, D. W., Bussian, E. R., ”A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups”, Coll. Math. J. 1992, 23(2), 150 - 152.
Malik, D. S., Mordeson, J. N., Sen, M. K., Introduction to Abstract Algebra, United States of America, 2007.
Wallis, W. D., A Beginner’s Guide to Graph Theory, Second Edition, Birkh¨auser : New York, 2006.
Wilson, R. J., Introduction to Graph Theory, Fourth Edition, Addison Wesley Longman Limited : England, 1996
Cayley, A., ”Desiderata and suggestions: No. 2. The Theory of groups: graphical representation”, Amer. J. Math 1878, 1(2), 174–176.
Abdollahi, A., Akbari, S., Maimani, H. R., ”Non-commuting graph of a group”, J. Algebra 2006, 298(2), 468-492.
Ma, X. L., Wei, H. Q., Yang, L. Y., ”The coprime graph of a group”, Int. J. Group Theory 2014, 3(3), 13–23.
Williams, J. S., ”The prime graph components of finite groups”, In The Santa Cruz Conference on Finite Groups (Univ California, Santa Cruz, Calif, 1979). Proc Sympos Pure Math 1981 (Vol. 37, pp. 195–196).
Al-Kaseasbeh, S., Erfanian, A., ”A bipartite graph associated to elements and cosets of subgroups of a finite group”, AIMS Mathematics 2021, 6(10), 10395–10404