Research Article
BibTex RIS Cite
Year 2022, , 248 - 255, 30.12.2022
https://doi.org/10.47000/tjmcs.885881

Abstract

References

  • Afkhami, M., Barati, Z., Khashyarmanesh, K., Planar zero-divisor graphs of partially ordered sets, Acta Math. Hungar., 137(2012), 27–35.
  • Afkhami, M., Khashyarmanesh, K., The comaximal graph of a lattice, Bull. Malays. Math. Sci. Soc., 37(1)(2014), 261–269.
  • Afkhami, M., Barati Z., Khashyarmanesh, K., A graph associated to a lattice , Ricerche Mat., 63(2014), 67–78.
  • Akbari, S., Alilou, A., Amjadi J., Sheikholeslami, S.M., The co-annihilating-ideal graphs of commutative rings, Canad. Math. Bull., 60(2017), 3–11.
  • Albu, T., Iosif, M., Modular C11 lattices and lattice preradicals, Journal of Algebra and Its Applications, 16(5)(2017), 1–19.
  • Anderson, D.F., Livingston P.S., The zero-divisor graph of commutative ring, J. Algebra, 217(1999), 434–447.
  • Beck, I., Coloring of commutative rings, J. Algebra, 116(1988), 208–226.
  • Beineke, L.W., Characterizations of derived graphs, J. Comb. Theory, 9(1970), 129–135.
  • Chartrand, G., Zhang, P., Chromatic Graph Theory, Chapman and Hall/CRC, 2008, 298p.
  • Chelvam, T.T., Nithya, S., A note on the zero divisor graph of a lattice, Trans. on Combin., 3(3)(2014), 51–59.
  • Curtis, A.R., Diesl, A.J., Rieck, J.C., Classifying annihilating-ideal graphs of commutative artinian rings, Communications in Algebra, 46(9)(2018), 4131–4147.
  • Davey, B.A., Priestly, H.A., Introduction to Lattices and Order, 2nd, Cambridge University Press, 2002, 312p.
  • Das, A.K., Nongsiang, D., On reduced zero-divisor graphs of posets, Journal of Discrete Mathematics, 2015(2015), 1–7.
  • Krithika, R., Mathew, R., Narayanaswamy, N.S., Sadagopan N., A Dirac-type characterization of k-chordal graphs, Discrete Math., 313(2013), 2865–2867.
  • Lu, D., Wu T., The Zero-divisor graphs of posets and an application to semigroups, Graphs and Combinatorics, 26(2010), 793–804.
  • Nimbhorkar, S., Deshmukh, D., The essential element graph of a lattice, Asian-European J of Math., 13(1)(2020), 1–9.
  • Nimbhokar, S.K., Wasadikar, M.P., Pawar, M.M., Coloring of lattices, Math. Slovaca, 60(2010), 419–434.
  • Nimbhokar, S.K., Vidya, S.D., Incomparability graphs of dismantable lattices , Asian-European J of Math., 13(1)(2020), 1–8.
  • Parsapour, A., Javaheri, K A., The embedding of annihilating-ideal graphs associated to lattices in the projective plane, Bull. Malays. Math. Sci. Soc., 42(2019), 1625–1638.
  • Wasadikar, M., Survase, P., Lattices, whose incomparability graphs have horns, J Discrete Algorithms, 23(2013), 63–75.
  • Xue, Z., Liu, L., Zero-divisor graphs of partially ordered sets, Appl. math. Lett., 23(2010), 449–452.

On the Essential Element Graph of a Lattice

Year 2022, , 248 - 255, 30.12.2022
https://doi.org/10.47000/tjmcs.885881

Abstract

Let $\mathcal{L}$ be a bounded lattice. The essential element graph of $\mathcal{L}$ is a simple undirected graph $\varepsilon_{\mathcal{L}}$ such that the elements $x,y$ of $\mathcal{L}$ form an edge in $\varepsilon_{\mathcal{L}}$, whenever $x \vee y $ is an essential element of $\mathcal{L}$. In this paper, we study properties of the essential elements of lattices and essential element graphs. We study the lattices whose zero-divisor graphs and incomparability graphs are isomorphic to its essential element graphs. Moreover, the line essential element graphs are investigated.

References

  • Afkhami, M., Barati, Z., Khashyarmanesh, K., Planar zero-divisor graphs of partially ordered sets, Acta Math. Hungar., 137(2012), 27–35.
  • Afkhami, M., Khashyarmanesh, K., The comaximal graph of a lattice, Bull. Malays. Math. Sci. Soc., 37(1)(2014), 261–269.
  • Afkhami, M., Barati Z., Khashyarmanesh, K., A graph associated to a lattice , Ricerche Mat., 63(2014), 67–78.
  • Akbari, S., Alilou, A., Amjadi J., Sheikholeslami, S.M., The co-annihilating-ideal graphs of commutative rings, Canad. Math. Bull., 60(2017), 3–11.
  • Albu, T., Iosif, M., Modular C11 lattices and lattice preradicals, Journal of Algebra and Its Applications, 16(5)(2017), 1–19.
  • Anderson, D.F., Livingston P.S., The zero-divisor graph of commutative ring, J. Algebra, 217(1999), 434–447.
  • Beck, I., Coloring of commutative rings, J. Algebra, 116(1988), 208–226.
  • Beineke, L.W., Characterizations of derived graphs, J. Comb. Theory, 9(1970), 129–135.
  • Chartrand, G., Zhang, P., Chromatic Graph Theory, Chapman and Hall/CRC, 2008, 298p.
  • Chelvam, T.T., Nithya, S., A note on the zero divisor graph of a lattice, Trans. on Combin., 3(3)(2014), 51–59.
  • Curtis, A.R., Diesl, A.J., Rieck, J.C., Classifying annihilating-ideal graphs of commutative artinian rings, Communications in Algebra, 46(9)(2018), 4131–4147.
  • Davey, B.A., Priestly, H.A., Introduction to Lattices and Order, 2nd, Cambridge University Press, 2002, 312p.
  • Das, A.K., Nongsiang, D., On reduced zero-divisor graphs of posets, Journal of Discrete Mathematics, 2015(2015), 1–7.
  • Krithika, R., Mathew, R., Narayanaswamy, N.S., Sadagopan N., A Dirac-type characterization of k-chordal graphs, Discrete Math., 313(2013), 2865–2867.
  • Lu, D., Wu T., The Zero-divisor graphs of posets and an application to semigroups, Graphs and Combinatorics, 26(2010), 793–804.
  • Nimbhorkar, S., Deshmukh, D., The essential element graph of a lattice, Asian-European J of Math., 13(1)(2020), 1–9.
  • Nimbhokar, S.K., Wasadikar, M.P., Pawar, M.M., Coloring of lattices, Math. Slovaca, 60(2010), 419–434.
  • Nimbhokar, S.K., Vidya, S.D., Incomparability graphs of dismantable lattices , Asian-European J of Math., 13(1)(2020), 1–8.
  • Parsapour, A., Javaheri, K A., The embedding of annihilating-ideal graphs associated to lattices in the projective plane, Bull. Malays. Math. Sci. Soc., 42(2019), 1625–1638.
  • Wasadikar, M., Survase, P., Lattices, whose incomparability graphs have horns, J Discrete Algorithms, 23(2013), 63–75.
  • Xue, Z., Liu, L., Zero-divisor graphs of partially ordered sets, Appl. math. Lett., 23(2010), 449–452.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Alper Ülker 0000-0001-5592-7450

Publication Date December 30, 2022
Published in Issue Year 2022

Cite

APA Ülker, A. (2022). On the Essential Element Graph of a Lattice. Turkish Journal of Mathematics and Computer Science, 14(2), 248-255. https://doi.org/10.47000/tjmcs.885881
AMA Ülker A. On the Essential Element Graph of a Lattice. TJMCS. December 2022;14(2):248-255. doi:10.47000/tjmcs.885881
Chicago Ülker, Alper. “On the Essential Element Graph of a Lattice”. Turkish Journal of Mathematics and Computer Science 14, no. 2 (December 2022): 248-55. https://doi.org/10.47000/tjmcs.885881.
EndNote Ülker A (December 1, 2022) On the Essential Element Graph of a Lattice. Turkish Journal of Mathematics and Computer Science 14 2 248–255.
IEEE A. Ülker, “On the Essential Element Graph of a Lattice”, TJMCS, vol. 14, no. 2, pp. 248–255, 2022, doi: 10.47000/tjmcs.885881.
ISNAD Ülker, Alper. “On the Essential Element Graph of a Lattice”. Turkish Journal of Mathematics and Computer Science 14/2 (December 2022), 248-255. https://doi.org/10.47000/tjmcs.885881.
JAMA Ülker A. On the Essential Element Graph of a Lattice. TJMCS. 2022;14:248–255.
MLA Ülker, Alper. “On the Essential Element Graph of a Lattice”. Turkish Journal of Mathematics and Computer Science, vol. 14, no. 2, 2022, pp. 248-55, doi:10.47000/tjmcs.885881.
Vancouver Ülker A. On the Essential Element Graph of a Lattice. TJMCS. 2022;14(2):248-55.