ORIGINAL_ARTICLE
The annihilator graph of a 0-distributive lattice
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.
http://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdf
2018-09-01T11:23:20
2018-02-22T11:23:20
1
18
10.22108/toc.2017.104919.1507
Distributive lattice
Annihilator graph
Zero-divisor graph
Saeid
Bagheri
bagheri_saeid@yahoo.com
true
1
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
LEAD_AUTHOR
Mahtab
Koohi Kerahroodi
mahtabkh3@gmail.com
true
2
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
AUTHOR
[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.
1
[2] S. Akbari and A. Mohammadian, On zero-divisor graph of finite rings, J. Algebra, 314 (2007) 168–184.
2
[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.
3
[4] D. D. Anderson and M. Naseer, Beck‘s coloring of a commutative ring, J. Algebra, 159 (1993) 500-514.
4
[5] D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434-447.
5
[6] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008) 2706–2719.
6
[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.
7
[8] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014) 108–121.
8
[9] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208–226.
9
[10] B. Bollobas and I. Rival, The maximal size of the covering graph of a lattice, Algebra Univ., 9 (1979) 371–373.
10
[11] J. Coykendal, S. Sather-Wagstaff, L. Sheppardson and S. Spiroff, On zero divisor graphs, Progress in commutative Algebra, 2 (2012) 241–299.
11
[12] F. R. Demeyer, T. Mckenzie and K. Schneider, The zero divisor graph of a commutative semigroup, Semigroup Forum, 65 (2002) 206–214.
12
[13] D. Duffus and I. Rival, Path length in the covering graph of a lattice, Discrete Math., 19 (1977) 139–158.
13
[14] S. Dutta and Ch. Lanong, On annihilator graphs of a finite commutative ring, Trans. Comb., 6 no. 1 (2017) 1-11.
14
[15] E. Estaji and K. Khashyarmanesh, The zero divisor graph of a lattice, Results Math., 61 (2012) 1–11.
15
[16] N. D. Filipov, Comparability graphs of partially ordered sets of different types, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 373–380.
16
[17] E. Gedeonova, Lattices whose covering graphs are S-graphs, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 407-435.
17
[18] G. Grätzer, Lattice Theory: Foundation, Birkhauser, Basel, 2011.
18
[19] V. Joshi, Zero divisor graphs of a poset with respect to an ideal, Order, 29 (2012) 499–506.
19
[20] V. Joshi and A. Khiste, On the zero divisor graphs of pm-lattices, Discrete Math., 312 (2012) 2076–2082.
20
[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.
21
[22] T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301 (2006) 174–193.
22
[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.
23
[24] Y. S. Pawar and N. K. Thakare, pm-lattices, Algebra Univ., 7 (1977) 259–263.
24
[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.
25
[26] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall Upper Saddle River, 2001.
26
[27] R. J. Wilson, Introduction to Graph Theory, Fourth edition, Longman, Harlow, 1996.
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ORIGINAL_ARTICLE
A spectral excess theorem for digraphs with normal Laplacian matrices
The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distance-regularity in graphs. Up to now, some authors have given some variations of this theorem. Motivated by this, we give the corresponding result by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distance-regular. Hence such a digraph is strongly regular with girth $g=2$ or $g=3$.
http://toc.ui.ac.ir/article_22346_7909f36dcf76d9cf3fc6a7dff2c384b9.pdf
2018-09-01T11:23:20
2018-02-22T11:23:20
19
27
10.22108/toc.2018.105873.1513
A Laplacian spectral excess theorem
Distance-regular digraphs
Strongly regular digraphs
Fateme
Shafiei
fatemeh.shafiei66@gmail.com
true
1
Isfahan University of Technology
Isfahan University of Technology
Isfahan University of Technology
LEAD_AUTHOR
[1] J. Bang-Jensen and G. Z. Gregory, Digraphs: Theory, Algorithms and Applications, Springer Monographs in
1
Mathematics, 2nd ed, 2009.
2
[2] A. E. Brouwer, personal homepage: http://www.cwi.nl/~aeb/math/dsrg/dsrg.html.
3
[3] A. E. Brouwer, A. M. Cohen and A. Neumaier, distance-regular Graphs, Springer-Verlag, Berlin-New York,
4
[4] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer, 2012; available online at http://homepages.cwi.nl/~aeb/math/ipm/.
5
[5] F. Comellas, M. A. Fiol, J. Gimbert and M. Mitjana, Weakly distance-regular digraphs, J. Combin. Theory
6
Ser. B, 90 (2004) 233–255.
7
[6] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Theory and Application, VEB Deutscher Verlag
8
der Wissenschaften, Berlin, second edition, 1982.
9
[7] C. Dalfo, E.R. Van Dam, M. A. Fiol and E. Garriga, Dual concepts of almost distance-regularity and the
10
spectral excess theorem, Discrete Math., 312 (2012) 2730–2734.
11
[8] C. Dalfo, E. R. Van Dam, M. A. Fiol, E. Garriga and B. L. Gorissen, On almost distance-regular graphs, J.
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Combin. Theory Ser. A, 118 (2011) 1094–1113.
13
[9] R. M. Damerell, Distance-transitive and distance-regular digraphs, J. Combin. Theory Ser. B, 31 (1981) 46–53.
14
[10] A. M. Duval, A directed graph version of strongly regular graphs, J. Combin. Theory Ser. A, 47 (1988) 71–100.
15
[11] E. R. Van Dam, The spectral excess theorem for distance-regular graphs: a global (over)view, Electron. J.
16
Combin., 15 (2008), pp. 10.
17
[12] E. R. Van Dam and M. A. Fiol, A short proof of the odd-girth theorem, Electron. J. Combin., 19 (2012) pp.
18
[13] E. R. Van Dam and M. A. Fiol, The Laplacian Spectral Excess Theprem for distance-Regular Graphs, Linear
19
Algebra Appl., 458 (2014) 1–6.
20
[14] Carl D. Meyer, Matrix analysis and applied linear algebra, Philadelphia, USA , 101 (2011) 486–489.
21
[15] M. A. Fiol, Algebraic characterizations of distance-regular graphs, Discrete Math., 246 (2002) 111–129.
22
[16] M. A. Fiol, On some approaches to the spectral excess theorem for nonregular graphs, J. Combin. Theory Ser. A, 120 (2013) 1285–1290.
23
[17] M. A. Fiol and E. Garriga, From local adjacency polynomials to locally pseudo-distance-regular graphs, J.
24
Combin. Theory Ser. B, 71 (1997) 162–183.
25
[18] M. A. Fiol, S. Gago and E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra Appl., 432 (2010) 2418–2422.
26
[19] M. A. Fiol, E. Garriga and J. L. A. Yebra, Locally pseudo-distance-regular graphs, J. Combin.Theory Ser. B,
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68 (1996) 179–205.
28
[20] G. S. Lee and C. W. Weng, The spectral excess theorem for general graphs, J. Combin. Theory Ser. A, 119 (2012) 1427–1431.
29
[21] G. R. Omidi, A spectral excess theorem for normal digraphs, J. Algebraic Combin., 42 (2015) 537–554.
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