LAPLACIAN SPECTRUM AND ENERGY OF NON-COMMUTING GRAPHS OF FINITE RINGS

Sharma, Monalisha; Nath, Rajat Kanti

doi:10.22044/jas.2024.14111.1802

LAPLACIAN SPECTRUM AND ENERGY OF NON-COMMUTING GRAPHS OF FINITE RINGS

Document Type : Original Manuscript

Authors

Monalisha Sharma ¹
Rajat Kanti Nath ²

¹ Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India. Department of Mathematics, Baosi Banikanta Kakati College Nagaon, Barpeta, PIN - 781311, Assam, India.

² Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India.

10.22044/jas.2024.14111.1802

Abstract

We compute spectrum, energy, Laplacian spectrum/ energy and signless Laplacian spectrum/energy of non-commuting graphs of certain finite non-commutative rings. In particular, we consider finite rings $R$ such that $|R| = p^2, p^3, p^4$, $p^5$, $p^2q$ and $p^3q$, where $p$ and $q$ are primes. Further, we consider $n$-centralizer finite\\ rings for $n \, = \,4, \, 5$ \, and \, $p \,+ \,2$; \, more generally, finite rings with central quotients isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$. Our computations reveal that non-commuting graphs of these rings are L-integral. We also determine whether non-commuting graphs of these rings are integral, Q-integral, hyperenergetic, L-hyperenergetic or Q-hyperenergetic.

Keywords

References

1. J. Dutta, D. K. Basnet and R. K. Nath, A note on n-centralizer finite rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), LXIV (2018), 161–171.

2. J. Dutta, D. K. Basnet and R. K. Nath, Characterizing some rings of finite order, Tamkang J. Math., 53 (2022), 97–108.

3. J. Dutta, D. K. Basnet and R. K. Nath, On generalized non-commuting graph of a finite ring, Algebra Colloq., 25 (2018), 149–160.

4. J. Dutta, W. N. T. Fasfous and R. K. Nath, Spectrum and genus of commuting graphs of some classes of finite rings, Acta Comment. Univ. Tartu. Math., 23 (2019), 5–12.

5. J. Dutta and R. K. Nath, Rings having four distinct centralizers, in Matrix, M. R. Publication, Assam, (2017), 12–18.

6. A. Erfanian, K. Khashyarmanesh and Kh. Nafar, Non-commuting graphs of rings, Discrete Math. Algorithms Appl., 7 (2015), Article ID: 1550027.

7. W. N. T. Fasfous, R. K. Nath and R. Sharafdini, Various spectra and energies of commuting graphs of finite rings, Hacet. J. Math. Stat., 49 (2020), 1915–1925.

8. I. Gutman, Hyperenergetic molecular graphs, J. Serb. Chem. Soc., 64 (1999), 199–205.

9. D. MacHale, Commutativity in finite rings, The American Mathematical Monthly, 83 (1976), 30–32.

10. B. Mohar, The Laplacian spectrum of graphs, in Graph Theory, Combinatorics, and Applications, Ed. Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk, Wiley, (1991), 871–898.

11. R. K. Nath, A note on super integral rings, Bol. Soc. Parana. Mat., 38 (2020), 213–218.

12. B. H. Neumann, A problem of Paul Erd¨os on groups, J. Aust. Math. Soc., 21 (1976), 467–472.

13. S. Pirzada, B. Rather, R. U. Shaban and T. Chishti, Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring ZpM1qM2, Commun. Comb. Optim., 8 (2023), 561–574.

14. S. Pirzada, B. Rather, R. U. Shaban and S. Merajuddin, On signless Laplacian spectrum of the zero divisor graph of the ring Zn, Korean J. Math., 29 (2021), 13–24.

1. J. Dutta, D. K. Basnet and R. K. Nath, A note on n-centralizer finite rings, An. Stiint.

Univ. Al. I. Cuza Iasi. Mat. (N.S.), LXIV (2018), 161–171.

2. J. Dutta, D. K. Basnet and R. K. Nath, Characterizing some rings of finite order,

Tamkang J. Math., 53 (2022), 97–108.

3. J. Dutta, D. K. Basnet and R. K. Nath, On generalized non-commuting graph of a finite

ring, Algebra Colloq., 25 (2018), 149–160.

4. J. Dutta, W. N. T. Fasfous and R. K. Nath, Spectrum and genus of commuting graphs

of some classes of finite rings, Acta Comment. Univ. Tartu. Math., 23 (2019), 5–12.

5. J. Dutta and R. K. Nath, Rings having four distinct centralizers, in Matrix, M. R.

Publication, Assam, (2017), 12–18.