Group-Annihilator Graphs Realised by Finite Abelian Groups and Its Properties

Springer Science and Business Media LLC - Tập 38 - Trang 1-24 - 2021
Eshita Mazumdar1, Rameez Raja2
1Mathematical and Physical Sciences, School of Arts and Sciences, Ahmedabad University, Ahmedabad, India
2Department of Mathematics, NIT Srinagar, Srinagar, India

Tóm tắt

Let G be a finite abelian group viewed a $${\mathbb {Z}}$$ -module and let $${\mathcal {G}} = (V, E)$$ be a simple graph. In this paper, we consider a graph $$\Gamma (G)$$ called as a group-annihilator graph. The vertices of $$\Gamma (G)$$ are all elements of G and two distinct vertices x and y are adjacent in $$\Gamma (G)$$ if and only if $$[x : G][y : G]G = \{0\}$$ , where $$x, y\in G$$ and $$[x : G] = \{r\in {\mathbb {Z}} : rG \subseteq {\mathbb {Z}}x\}$$ is an ideal of a ring $${\mathbb {Z}}$$ . We discuss in detail the graph structure realised by a group G. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annihilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: $$Aut(\Gamma (G)) \times G \rightarrow G$$ .

Tài liệu tham khảo

Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999) Bapat, R.B.: Graphs and matrices. Springer/Hindustan Book Agency, London (2010) Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988) Bohdan, Z.: Intersection graphs of finite abelian groups. Czechoslov. Math. J. 25(2), 171–174 (1975) Brauer, R., Fowler, K.A.: On groups of even order. Ann. Math. 62(2), 565–583 (1955) Cameron, P., Ghosh, S.: The power graph of a finite group. Discret. Math. 311(13), 1220–1222 (2011) Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Ann. Discret. Math. 1, 145–162 (1977) Chung, F.: Spectral graph theory. AMS 92 (1997) Hammer, P.L., Kelmans, A.K.: Laplacian spectra and spanning trees of threshold graphs. Discret. Appl. Math. 65, 255–273 (1996) Henderson, P.B., Zalcstein, Y.: A graph-theoretic characterization of the PV class of synchronizing primitives. SIAM J. Comput. 6(1), 88–108 (1977) Liebeck, M.W., Shalev, A.: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky. J. Algebra 184, 31–57 (1996) Merris, R.: Graph theory. John Wiley and Sons, Hoboken (2011) Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197, 143–176 (1994) Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326, 1472–1475 (2007) Dutta, K., Prasad, A.: Degenerations and orbits in finite abelian groups. J. Comb. Theory Ser. A 118, 1685–1694 (2011) Gutman, I.: Hyperenergetic and hypoenergetic graphs. Zbornik Radova 22, 113–135 (2011) Jacobs, D.P., Trevisan, V., Tura, F.: Eigenvalues and energy in threshold graphs. Linear Algebra Appl. 465, 412–425 (2015) Jacobs, D.P., Trevisan, V., Tura, F.: Computing the characteristic polynomial of threshold graphs. J. Graph Algorithms Appl. 18(5), 709–719 (2014) Mahadev, N.V.R., Peled, U.N.: Threshold graphs and related topics. Ann. Discret. Math. 56 (1995) Miller, G.A.: Determination of all the characteristic subgroups of any abelian group. Am. J. Math. 27(1), 15–24 (1905) Pirzada, S., Raja, R., Redmond, S.P.: Locating sets and numbers of graphs associated to commutative rings. J. Algebra Appl. 13(7), 1450047 (2014) Pirzada, S., Raja, R.: On the metric dimension of a zero-divisor graph. Commun. Algebra 45(4), 1399–1408 (2017) Raja, R., Pirzada, S., Redmond, S.P.: On Locating numbers and codes of zero-divisor graphs associated with commutative rings. J. Algebra Appl. 15(1), 1650014 (2016) Raja, R., Pirzada, S.: On annihilating graphs of modules over commutative rings. Algebra Colloq. (To appear) Redmond, S.P.: An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra 31, 4425–4443 (2003) Schwachhöfer, M., Stroppel, M.: Finding representatives for the orbits under the automorphism group of a bounded abelian group. J. Algebra 211(1), 225–239 (1999)