Bagheri, S., Koohi Kerahroodi, M. (2018). The annihilator graph of a 0-distributive lattice. Transactions on Combinatorics, 7(3), 1-18. doi: 10.22108/toc.2017.104919.1507

Saeid Bagheri; Mahtab Koohi Kerahroodi. "The annihilator graph of a 0-distributive lattice". Transactions on Combinatorics, 7, 3, 2018, 1-18. doi: 10.22108/toc.2017.104919.1507

Bagheri, S., Koohi Kerahroodi, M. (2018). 'The annihilator graph of a 0-distributive lattice', Transactions on Combinatorics, 7(3), pp. 1-18. doi: 10.22108/toc.2017.104919.1507

Bagheri, S., Koohi Kerahroodi, M. The annihilator graph of a 0-distributive lattice. Transactions on Combinatorics, 2018; 7(3): 1-18. doi: 10.22108/toc.2017.104919.1507

^{1}Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran

^{2}Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.

Abstract

In this article, for a lattice $\mathcal L$, we define and investigate the annihilator graph $\mathfrak {ag} (\mathcal L)$ of $\mathcal L$ which contains the zero-divisor graph of $\mathcal L$ as a subgraph. Also, for a 0-distributive lattice $\mathcal L$, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice $\mathcal L$ with $Z(\mathcal L)\neq\lbrace 0\rbrace$, we show that $\mathfrak {ag} (\mathcal L) = \Gamma(\mathcal L)$ if and only if $\mathcal L$ has exactly two minimal prime ideals. Among other things, we consider the annihilator graph $\mathfrak {ag} (\mathcal L)$ of the lattice $\mathcal L=(\mathcal D(n),|)$ containing all positive divisors of a non-prime natural number $n$ and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which $\mathfrak {ag} (\mathcal D(n))$ or $\Gamma(\mathcal D(n))$ are planar, Eulerian or Hamiltonian.

[1] M. Afkhami, K. Khashyarmanesh and Z. Rajabi, Some results on the annihilator graph of a commutative ring, Czechoslovak Math. Journal, 67 (2017) 151–169.

[2] S. Akbari and A. Mohammadian, On zero-divisor graph of finite rings, J. Algebra, 314 (2007) 168–184.

[3] M. Alizadeh, A. K. Das, H. R. Maimani, M. R. Pournaki and S. Yassemi, On the diameter and girth of zero-divisor graphs of posets, Discrete Appl. Math., 160 (2012) 1319–1324.

[4] D. D. Anderson and M. Naseer, Beck‘s coloring of a commutative ring, J. Algebra, 159 (1993) 500-514.

[5] D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434-447.

[6] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008) 2706–2719.

[7] D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007) 543–550.

[8] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014) 108–121.

[9] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208–226.

[10] B. Bollobas and I. Rival, The maximal size of the covering graph of a lattice, Algebra Univ., 9 (1979) 371–373.

[11] J. Coykendal, S. Sather-Wagstaff, L. Sheppardson and S. Spiroff, On zero divisor graphs, Progress in commutative Algebra, 2 (2012) 241–299.

[12] F. R. Demeyer, T. Mckenzie and K. Schneider, The zero divisor graph of a commutative semigroup, Semigroup Forum, 65 (2002) 206–214.

[13] D. Duffus and I. Rival, Path length in the covering graph of a lattice, Discrete Math., 19 (1977) 139–158.

[14] S. Dutta and Ch. Lanong, On annihilator graphs of a finite commutative ring, Trans. Comb., 6 no. 1 (2017) 1-11.

[15] E. Estaji and K. Khashyarmanesh, The zero divisor graph of a lattice, Results Math., 61 (2012) 1–11.

[16] N. D. Filipov, Comparability graphs of partially ordered sets of different types, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 373–380.

[17] E. Gedeonova, Lattices whose covering graphs are S-graphs, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 407-435.

[18] G. Grätzer, Lattice Theory: Foundation, Birkhauser, Basel, 2011.

[19] V. Joshi, Zero divisor graphs of a poset with respect to an ideal, Order, 29 (2012) 499–506.

[20] V. Joshi and A. Khiste, On the zero divisor graphs of pm-lattices, Discrete Math., 312 (2012) 2076–2082.

[21] V. Joshi and S. Sarode, Diameter and girth of zero divisor graph of multiplicative lattices, Asian-Eur. J. Math., 9 (2016). http://dx.doi.org/10.1142/S1793557116500716.

[22] T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301 (2006) 174–193.

[23] M. J. Nikmehr, R. Nikandish and M. Bakhtyiari, More on the annihilator graph of a commutative ring, Hokkaido Math. J., 46 (2017) 107–118.

[24] Y. S. Pawar and N. K. Thakare, pm-lattices, Algebra Univ., 7 (1977) 259–263.

[25] T. Tamizh Chelvam and S. Nithya, A note on the zero divisor graph of a lattice, Trans. Comb., 3 no. 3 (2014) 51–59.

[26] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall Upper Saddle River, 2001.

[27] R. J. Wilson, Introduction to Graph Theory, Fourth edition, Longman, Harlow, 1996.