# On the connectivity of the annihilating-ideal graphs

• Volume: 35, Issue: 2, page 195-204
• ISSN: 1509-9415

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## Abstract

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Let R be a commutative ring with identity and 𝔸*(R) the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸*(R) and two distinct vertices I₁ and I₂ are adjacent if and only if I₁I₂ = (0). In this paper, we examine the presence of cut vertices and cut sets in the annihilating-ideal graph of a commutative Artinian ring and provide a partial classification of the rings in which they appear. Using this, we obtain the vertex connectivity of some annihilating-ideal graphs.

## How to cite

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T. Tamizh Chelvam, and K. Selvakumar. "On the connectivity of the annihilating-ideal graphs." Discussiones Mathematicae - General Algebra and Applications 35.2 (2015): 195-204. <http://eudml.org/doc/276649>.

@article{T2015,
abstract = {Let R be a commutative ring with identity and 𝔸*(R) the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸*(R) and two distinct vertices I₁ and I₂ are adjacent if and only if I₁I₂ = (0). In this paper, we examine the presence of cut vertices and cut sets in the annihilating-ideal graph of a commutative Artinian ring and provide a partial classification of the rings in which they appear. Using this, we obtain the vertex connectivity of some annihilating-ideal graphs.},
author = {T. Tamizh Chelvam, K. Selvakumar},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {annihilating-ideal graph; local ring; nilpotency; cut vertex},
language = {eng},
number = {2},
pages = {195-204},
title = {On the connectivity of the annihilating-ideal graphs},
url = {http://eudml.org/doc/276649},
volume = {35},
year = {2015},
}

TY - JOUR
AU - T. Tamizh Chelvam
AU - K. Selvakumar
TI - On the connectivity of the annihilating-ideal graphs
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2015
VL - 35
IS - 2
SP - 195
EP - 204
AB - Let R be a commutative ring with identity and 𝔸*(R) the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸*(R) and two distinct vertices I₁ and I₂ are adjacent if and only if I₁I₂ = (0). In this paper, we examine the presence of cut vertices and cut sets in the annihilating-ideal graph of a commutative Artinian ring and provide a partial classification of the rings in which they appear. Using this, we obtain the vertex connectivity of some annihilating-ideal graphs.
LA - eng
KW - annihilating-ideal graph; local ring; nilpotency; cut vertex
UR - http://eudml.org/doc/276649
ER -

## References

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1. [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. doi: doi:10.1006/jabr.1998.7840 Zbl0941.05062
2. [2] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company (1969).
3. [3] M. Axtell, N. Baeth and J. Stickles, Cut vertices in zero-divisor graph of finite commutative rings, Comm. Algebra 39 (2011), 2179-2188. doi: 10.1080/00927872.2010.488681 Zbl1226.13007
4. [4] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (4) (2011), 727-739. doi: 10.1142/S0219498811004896 Zbl1276.13002
5. [5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), 741-753. doi: 10.1142/S0219498811004902 Zbl1276.13003
6. [6] G. Chartrand and L. Lesniak, Graphs and Digraphs (Wadsworth and Brooks/Cole, Monterey, CA (1986).
7. [7] B. Cote, C. Ewing, M. Huhn, C.M. Plaut and D. Weber, Cut sets in zero-divisor graphs of finite commutative rings, Comm. Algebra 39 (2011), 2849-2861. doi: 10.1080/00927872.2010.489534 Zbl1228.13011
8. [8] I. Kaplansky, Commutative Rings, rev. ed. University of Chicago Press Chicago (1974). Zbl0296.13001
9. [9] S.P. Redmond, Central sets and radii of the zero-divisor graphs of commutative rings, Comm. Algebra 34 (2006), 2389-2401. doi: 10.1080/00927870600649103 Zbl1105.13007
10. [10] T. Tamizh Chelvam and K. Selvakumar, Central sets in the annihilating-ideal graph of commutative rings, J. Combin. Math. Combin. Comput. 88 (2014), 277-288. Zbl1293.05156
11. [11] A.T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam (1973).

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