Laplacian and Wiener index of extension of zero divisor graph. (English) Zbl 07894681
Summary: The main purpose of this paper is to study the Laplacian eigenvalues of the extension of the zero divisor graph, \( \Gamma_e ( \mathbb{Z}_n ) ,\) for some particular values of \(n\). We characterize the values of \(n\) that give the equality of the spectral radius and the second-smallest eigenvalue of \(\Gamma_e ( \mathbb{Z}_n )\). Finding Wiener index of \(\Gamma_e ( \mathbb{Z}_n )\) in terms of its Laplacian eigenvalues is another objective of this paper.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C12 | Distance in graphs |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 13A70 | General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.) |
References:
| [1] | Anderson, D. F.; Badawi, A., The total graph of a commutative ring, J. Algebra, 320, 2706-2719, 2008 · Zbl 1158.13001 |
| [2] | Anderson, D. F.; Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra, 217, 434-447, 1999 · Zbl 0941.05062 |
| [3] | Asir, T.; Rabikka, V., The Wiener index of the zero- divisor graph of \(\mathbb{Z}_n\), Discrete Appl. Math., 319, 461-471, 2022 · Zbl 1494.05025 |
| [4] | Bapat, R. B., Graph and Matrices, 2019, Hindustan Book Agency: Hindustan Book Agency New Delhi |
| [5] | Bapat, R. B.; Lal, A. K.; Pati, S., Some observations on algebraic connectivity of graphs, Appl. Linear Algebra, Probab. Stat., 177-190, 2023 |
| [6] | Beck, I., Coloring of commutative ring, J. Algebra, 208-226, 1988 · Zbl 0654.13001 |
| [7] | P. Bora, K.K. Rajkhowa, Laplacian spectrum of a subgraph of the total graph of \(Z_n\). communicated. |
| [8] | Chattopadhyay, S.; Patra, K. L.; Sahoo, B. K., Laplacian eigenvalues of the zero divisor graph of the ring \(Z_n\), Linear Algebra Appl., 584, 267-286, 2020 · Zbl 1426.05062 |
| [9] | Cherrabi, A.; Essannouni, H.; Jabbouri, E.; Ouadfel, A., On a new extension of the zero- divisor graph, Algebra Colloq., 27, 03, 469-476, 2020 · Zbl 1442.13019 |
| [10] | Mohar, B., The Laplacian spectrum of graphs, (Graph Theory, Combinatorics, and Applications, vol. 2, 1991, Wiley-Intersci. Publ., Wiley: Wiley-Intersci. Publ., Wiley New York), 871-898, (Kalamazoo, MI, 1988) · Zbl 0840.05059 |
| [11] | Rashid, M.; Amal, S. A.; Wasim, A.; Muzibur, R. M., Spectrum of the cozero- divisor graph associated to ring \(\mathbb{Z}_n\), MDPI, 957, 2023 |
| [12] | Rashid, M.; Mozumder, M. R.; Anwar, M., Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring \(\mathbb{Z}_n\), Georgian Math. J., 2023 |
| [13] | Young, M., Adjacency matrices of zero- divisor graphs of integer modulo \(n\), Involve, 8, 753-761, 2015 · Zbl 1322.05093 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
