Araştırma Makalesi
Yıl 2025,
Cilt: 54 Sayı: 3, 939 - 957, 24.06.2025
Öz
Let $R$ be a finite commutative ring with nonzero identity. Let $Z^{*}(R)$ be the set of nonzero zero-divisors of $R$. We are dealing with the zero-divisor graph of $R$ which is denoted by $\Gamma(R)$ with vertex set $Z^{*}(R)$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. The motivation of this study is to compute Wiener index in algebraic graph theory for special type of graph called zero-divisor graph. Wiener index is defined as the sum of all distance between all pairs of vertices in $\Gamma(R)$. In addition, we generalize the Wiener index of the zero-divisor graph in $\mathbb{Z}_p[x]/(x^2)$ for any prime number $p$. We obtain our results and methods by tables and figures.
Anahtar Kelimeler
local Rings zero-divisor graph compressed zero-divisor graph Wiener index
Kaynakça
- [1] MR. Ahmadi and R. Jahani-Nezhad, Energy and Wiener index of zero divisor graphs, J. Iranian of Math Chem. 2 (1), 45-51, 2011.
- [2] D. F. Anderson, T. Asir, A. Badawi, and T. T. Chelvam, Graphs from rings, Springer, 2021.
- [3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217, No. 2, 434- 447, 1999.
- [4] T. Asir, A. Kumar and A. Mehdi, On the zero-divisor hypergraph of a reduced ring, Acta Math. Hungar. 1-14, 2023.
- [5] T. Asir, V. Rabikka, A. M. Anto and N. Shunmugapriya, Wiener index of graphs over rings: a survey, AKCE International Journal of Graphs and Combinatorics 19(3), 316-324, 2022.
- [6] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley Publishing, Reading, MA, 1969.
- [7] I. Beck, Coloring of commutative rings, J. Algebra 116, 208226, 1988.
- [8] D. Bonchev, The Wiener numbersome applications and new developments, In Topology in Chem, 58-88, 2002.
- [9] A. Cayley, Desiderata and suggestions: The theory of groups: Graphical representation, Amer. J. Math. 1, 174176, 1878.
- [10] A. Dobrynin, R. Entringer and I. N Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 3, 211-249, 2001.
- [11] J. B. Fraleigh, A first course in abstract algebra, Pearson Education India 2003.
- [12] A. Graovac and T. Pisanski, On the Wiener index of a graph, J. math chem. 8 (1), 53-62, 1991.
- [13] M. Hazewinkel, Formal groups and applications, Vol. 78. Elsevier, 1978.
- [14] H. Hosoya, Topological index. a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (9), 23322339, 1971.
- [15] H. M. Hoque, H. K. Saikia, J. Goswami and D. Patwari, Non-Nilpotent Graph of Commutative Rings, J. Algebraic System 12 (1), 149-162, 2024.
- [16] T. W. Hungerford, Algebra, Springer Science and Business Media, 2003.
- [17] W. Hungerford, Thomas Abstract algebra: an introduction, Cengage Learning, 2012.
- [18] D. Janei, A. Milievi, S. Nikoli, N. Trinajsti, Graph-theoretical matrices in chemistry, Second edition, CRC Press, Boca Raton. F. xiv+160 pp. ISBN: 978-1-4987-0115-0, 2015.
- [19] D. J. Klein, I Lukovits, I Gutman, On the defination of the hyper Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35, 50-52, 1995.
- [20] B. Külshammer, Lectures on block theory, Cambridge University Press, 1991.
- [21] J.J. Molitierno, Applications of combinatorial matrix theory to Laplacian matrices of graphs, CRC Press, 2016.
- [22] GR. Omidi and E. Vatandoost, On the commuting graph of rings, J. Algebra Appl. 10 (3), 521-52, 2011.
- [23] S. Pirzada, An Introduction to Graph Theory, Universities Press, Hyderabad, India, 2011.
- [24] Q. Ma, Extremal polygonal chains with respect to the Kirchhoff index, Disc. Appl. Math. 342, 218-226, 2024.
- [25] M. Randitc, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211, 478-483, 1993.
- [26] N. U. Rehman, M. Nazim and K. Selvakumar, On the genus of extended zero-divisor graph of commutative rings, Rend. Circ. Mat. Palermo 2, 1-10, 2022.
- [27] K. Selvakumar, P. Gangaeswari and G. Arunkumar, The Wiener index of the zerodivisor graph of a finite commutative ring with unity, Dis. Appl. Math. 311, 72-84, 2022.
- [28] NH. Shuker and PA. Rashed, The zero divisor graph of the ring Zpqr, Int. J. Sci. 6 (2), 569-574, 2015.
- [29] H. Wiener, Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 69, 17-20, 1947.
- [30] R. Wilson. J, Introduction to graph theory. Pearson Education India, 1979.
- [31] K. Xu, M. Liu, K. C. Das, I. Gutman, and B. Furtula, A survey on graphs extremal with respect to distance-based topologicalindices, MATCH Commun. Math. Comput. Chem, 71 (3), 461508, 2014.
- [32] K. Yamazaki, M. Qian and R. Uehara, Efficient enumeration of non-isomorphic distance-hereditary graphs and related graphs, Disc. App. Math. 342, 190-199, 2024.
- [33] Y.-N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Disc. Math, 135, 359365, 1994.
Yıl 2025,
Cilt: 54 Sayı: 3, 939 - 957, 24.06.2025
Öz
Kaynakça
- [1] MR. Ahmadi and R. Jahani-Nezhad, Energy and Wiener index of zero divisor graphs, J. Iranian of Math Chem. 2 (1), 45-51, 2011.
- [2] D. F. Anderson, T. Asir, A. Badawi, and T. T. Chelvam, Graphs from rings, Springer, 2021.
- [3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217, No. 2, 434- 447, 1999.
- [4] T. Asir, A. Kumar and A. Mehdi, On the zero-divisor hypergraph of a reduced ring, Acta Math. Hungar. 1-14, 2023.
- [5] T. Asir, V. Rabikka, A. M. Anto and N. Shunmugapriya, Wiener index of graphs over rings: a survey, AKCE International Journal of Graphs and Combinatorics 19(3), 316-324, 2022.
- [6] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley Publishing, Reading, MA, 1969.
- [7] I. Beck, Coloring of commutative rings, J. Algebra 116, 208226, 1988.
- [8] D. Bonchev, The Wiener numbersome applications and new developments, In Topology in Chem, 58-88, 2002.
- [9] A. Cayley, Desiderata and suggestions: The theory of groups: Graphical representation, Amer. J. Math. 1, 174176, 1878.
- [10] A. Dobrynin, R. Entringer and I. N Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 3, 211-249, 2001.
- [11] J. B. Fraleigh, A first course in abstract algebra, Pearson Education India 2003.
- [12] A. Graovac and T. Pisanski, On the Wiener index of a graph, J. math chem. 8 (1), 53-62, 1991.
- [13] M. Hazewinkel, Formal groups and applications, Vol. 78. Elsevier, 1978.
- [14] H. Hosoya, Topological index. a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (9), 23322339, 1971.
- [15] H. M. Hoque, H. K. Saikia, J. Goswami and D. Patwari, Non-Nilpotent Graph of Commutative Rings, J. Algebraic System 12 (1), 149-162, 2024.
- [16] T. W. Hungerford, Algebra, Springer Science and Business Media, 2003.
- [17] W. Hungerford, Thomas Abstract algebra: an introduction, Cengage Learning, 2012.
- [18] D. Janei, A. Milievi, S. Nikoli, N. Trinajsti, Graph-theoretical matrices in chemistry, Second edition, CRC Press, Boca Raton. F. xiv+160 pp. ISBN: 978-1-4987-0115-0, 2015.
- [19] D. J. Klein, I Lukovits, I Gutman, On the defination of the hyper Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35, 50-52, 1995.
- [20] B. Külshammer, Lectures on block theory, Cambridge University Press, 1991.
- [21] J.J. Molitierno, Applications of combinatorial matrix theory to Laplacian matrices of graphs, CRC Press, 2016.
- [22] GR. Omidi and E. Vatandoost, On the commuting graph of rings, J. Algebra Appl. 10 (3), 521-52, 2011.
- [23] S. Pirzada, An Introduction to Graph Theory, Universities Press, Hyderabad, India, 2011.
- [24] Q. Ma, Extremal polygonal chains with respect to the Kirchhoff index, Disc. Appl. Math. 342, 218-226, 2024.
- [25] M. Randitc, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211, 478-483, 1993.
- [26] N. U. Rehman, M. Nazim and K. Selvakumar, On the genus of extended zero-divisor graph of commutative rings, Rend. Circ. Mat. Palermo 2, 1-10, 2022.
- [27] K. Selvakumar, P. Gangaeswari and G. Arunkumar, The Wiener index of the zerodivisor graph of a finite commutative ring with unity, Dis. Appl. Math. 311, 72-84, 2022.
- [28] NH. Shuker and PA. Rashed, The zero divisor graph of the ring Zpqr, Int. J. Sci. 6 (2), 569-574, 2015.
- [29] H. Wiener, Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 69, 17-20, 1947.
- [30] R. Wilson. J, Introduction to graph theory. Pearson Education India, 1979.
- [31] K. Xu, M. Liu, K. C. Das, I. Gutman, and B. Furtula, A survey on graphs extremal with respect to distance-based topologicalindices, MATCH Commun. Math. Comput. Chem, 71 (3), 461508, 2014.
- [32] K. Yamazaki, M. Qian and R. Uehara, Efficient enumeration of non-isomorphic distance-hereditary graphs and related graphs, Disc. App. Math. 342, 190-199, 2024.
- [33] Y.-N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Disc. Math, 135, 359365, 1994.
Toplam 33 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 11 Nisan 2025 |
| Yayımlanma Tarihi | 24 Haziran 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 54 Sayı: 3 |