Abstract
Once the spectral radius and energy of a graph structure have been defined, many properties have been studied. The spectral radius and energy of a graph are related to the eigenvalues of the adjacency matrix of the graph. In this paper, we define an adjacency matrix for a prime ideal sum ($PIS$) graph and then extend the concepts of spectral radius and energy to $PIS$ graphs. Some bound theorems on the energy and spectral radius of $PIS$ graph structures are given. A SageMath code for plotting these graphs is also
provided.
References
- Bondy, J., & Murty, U. (1982). Graph theory with applications. Elsevier Science Publishing.
- Hogben, L. (2005). Spectral graph theory and the inverse eigenvalue problem of a graph. The Electronic Journal of Linear Algebra, 14, 12–31. https://doi.org/10.13001/1081-3810.1174
- Bapat, R. (2013). On the adjacency matrix of a threshold graph. Linear Algebra and its Applications, 439(10), 3008–3015. https://doi.org/10.1016/j.laa.2013.08.007
- Das, K., & Kumar, P. (2004). Some new bounds on the spectral radius of graphs. Discrete Mathematics, 281(1-3), 149–161. https://doi.org/10.1016/j.disc.2003.08.005
- Gutman, I. (1978). The energy of a graph. Ber Math— Statist Sekt Forschungsz Graz, 103, 1–22.
- Anderson, D., & Livingston, P. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217, 434–447. https://doi.org/10.1006/jabr.1998.7840
- Beck, I. (1988). Coloring of commutative rings. Journal of Algebra, 116(1), 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
- Banerjee, S. (2022). Laplacian spectrum of comaximal graph of the ring Zn. Journal of Algebra, 10(1), 285–298. https://doi.org/10.48550/arXiv.2005.02316
- Fasfous, W., Rajat, K., & Sharafdini, R. (2020). Various spectra and energies of commuting graphs of finite rings. Hacettepe Journal of Mathematics and Statistics, 49(6), 1915–1925. https://doi.org/10.15672/hujms.540309
- Sözen, E., Eryaşar, E., & Abdioğlu, C. (2022). Forgotten topological and wiener indices of prime ideal sum graph of Zn. Current Organic Synthesis. https://doi.org/10.2174/1570179420666230606140448
- Saha, M., Çelikel, E., & Abdioğlu, C. (2023). Prime ideal sum graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics, 22(06), 2350121–.https://doi.org/10.1142/S0219498823501219
Abstract
Bir graf yapısının spektral yarıçapı ve enerjisi tanımlandıktan sonra birçok özelliği incelenmiştir. Grafların spektral yarıçapı ve enerjisi komşuluk matrisin özdeğerleriyle ilişkilidir. Bu çalışmada bir asal ideal toplam
($PIS$) graf için bir komşuluk matrisi tanımlanmıştır ve daha sonra spektral yarıçap ve enerji kavramları $PIS$ grafları için genişletilmiştir. $PIS$ graf yapılarının enerjisi ve spektral yarıçapına ilişkin bazı sınır teoremleri verilmiştir. Ayrıca bu grafları çizmek için bir SageMath kodu da sunulmaktadır.
References
- Bondy, J., & Murty, U. (1982). Graph theory with applications. Elsevier Science Publishing.
- Hogben, L. (2005). Spectral graph theory and the inverse eigenvalue problem of a graph. The Electronic Journal of Linear Algebra, 14, 12–31. https://doi.org/10.13001/1081-3810.1174
- Bapat, R. (2013). On the adjacency matrix of a threshold graph. Linear Algebra and its Applications, 439(10), 3008–3015. https://doi.org/10.1016/j.laa.2013.08.007
- Das, K., & Kumar, P. (2004). Some new bounds on the spectral radius of graphs. Discrete Mathematics, 281(1-3), 149–161. https://doi.org/10.1016/j.disc.2003.08.005
- Gutman, I. (1978). The energy of a graph. Ber Math— Statist Sekt Forschungsz Graz, 103, 1–22.
- Anderson, D., & Livingston, P. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217, 434–447. https://doi.org/10.1006/jabr.1998.7840
- Beck, I. (1988). Coloring of commutative rings. Journal of Algebra, 116(1), 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
- Banerjee, S. (2022). Laplacian spectrum of comaximal graph of the ring Zn. Journal of Algebra, 10(1), 285–298. https://doi.org/10.48550/arXiv.2005.02316
- Fasfous, W., Rajat, K., & Sharafdini, R. (2020). Various spectra and energies of commuting graphs of finite rings. Hacettepe Journal of Mathematics and Statistics, 49(6), 1915–1925. https://doi.org/10.15672/hujms.540309
- Sözen, E., Eryaşar, E., & Abdioğlu, C. (2022). Forgotten topological and wiener indices of prime ideal sum graph of Zn. Current Organic Synthesis. https://doi.org/10.2174/1570179420666230606140448
- Saha, M., Çelikel, E., & Abdioğlu, C. (2023). Prime ideal sum graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics, 22(06), 2350121–.https://doi.org/10.1142/S0219498823501219
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 29, 2024 |
| Submission Date | August 14, 2023 |
| Published in Issue | Year 2024 Volume: 9 Issue: 1 |
