Spectral Properties of Identity Graph for Group of Integers Modulo using Degree-Based Matrices
Resumen
This paper investigates the spectral properties of the identity graph associated with the group $\mathbb{Z}_n$, utilizing five degree-based matrices. Specifically, the study employs the maximum and minimum degree, greatest common divisor, and first and second Zagreb matrices. For each case, the characteristic polynomial and the corresponding graph energy are derived. Furthermore, a comparative analysis is conducted between the computed energies and established results in the literature.
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\bibitem{Cameron} Cameron, P. J., {\it What can graphs and algebraic structures say to each other?}. AKCE International Journal of Graphs and Combinatorics 00, 1--6, (2023).
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\bibitem{Kumari} Kumari, L. M. and Pandiselvi, L. and Palani, K., {\it Quotient wnergy of zero divisor graphs and identity graph}. Baghdad Science of Journal 20(1(SI)), 277--282, (2023).
\bibitem{Li} Li, X. and Shi, Y. and Gutman, I., {\it Graph Energy}. New York, Springer (2012).
\bibitem{Rad} Rad, N. J. and Jahanbani, A. and Gutman, I., {\it Zagreb energy and Zagreb estrada index of graphs}. MATCH Communications in Mathematical and in Computer Chemistry 201879, 371--386, (2018).
\bibitem{Ramkumar} Ramkumar, R. and Nagarajan, K., {\it Greatest common divisor degree energy of graphs}. International Journal of Mathematical Sciences and Engineering Applications 11(II), 163--171, (2017).
\bibitem{Ramkumar} Ramkumar, R. and Nagarajan, K., {\it Greatest common divisor degree energy of graphs}. International Journal of Mathematical Sciences and Engineering Applications 11(II), 163--171, (2017).
\bibitem{Rilwan} Rilwan, N. M. and Hussain, A. S., {\it Different kinds of cordial labeling on identity graphs}. European Chemical Bulletin 12(10), 12374--12381, (2023).
\bibitem{Romdhini1} Romdhini, M. U. and Nawawi, A. and Syechah, B. N., {\it Seidel Laplacian and Seidel signless Laplacian energies of commuting graph for dihedral groups}. Malaysian Journal of Fundamental and Applied Sciences 20(3), 701--713, (2024).
\bibitem{Romdhini2} Romdhini, M. U. and Nawawi, A. and Al-Sharqi, F. and Al-Quran, A., {\it Characteristic polynomial of power graph for dihedral groups using degree-based matrices}. Malaysian Journal of Fundamental and Applied Sciences 20(2), 328--335, (2024).
\bibitem{Romdhini3} Romdhini, M. U. and Nawawi, A., {\it Maximum and minimum degree energy of commuting graph for dihedral groups}. Sains Malaysiana 51(12), 4145--4151, (2022).
\bibitem{Romdhini7} Romdhini, M. U. and Nawawi, A. and Al-Sharqi, F. and Salwa, {\it On Spectrum and Energy of Identity Graph for Group of Integers Modulo $n$, $\mathbb{Z}_n$}. European Journal of Pure and Applied Mathematics 17(4), 2915--2929, (2024).
\bibitem{Adiga2010} Adiga, C. and Swamy, S., {\it Bounds on the largest of minimum degree eigenvalues of graphs}. International Mathematics Forum 5(37), 1823--1831, (2010).
\bibitem{Bapat2004} Bapat, R. B. and Pati, S., {\it Energy of a graph is never an odd integer}. Bulletin of Kerala Mathematics Association 1, 129--132, (2004).
\bibitem{Cameron} Cameron, P. J., {\it What can graphs and algebraic structures say to each other?}. AKCE International Journal of Graphs and Combinatorics 00, 1--6, (2023).
\bibitem{Gutman} Gutman, I., {\it The energy of graph}. Ber. Math.-Stat. Sekt. Forschungsz. Graz. 102, 1--2, (1978).
\bibitem{Kandasamy} Kandasamy, W. B. V. and Smarandache, F., {\it Groups as Graphs}. Romania, Editura CuArt (2009).
\bibitem{Kumari} Kumari, L. M. and Pandiselvi, L. and Palani, K., {\it Quotient wnergy of zero divisor graphs and identity graph}. Baghdad Science of Journal 20(1(SI)), 277--282, (2023).
\bibitem{Li} Li, X. and Shi, Y. and Gutman, I., {\it Graph Energy}. New York, Springer (2012).
\bibitem{Rad} Rad, N. J. and Jahanbani, A. and Gutman, I., {\it Zagreb energy and Zagreb estrada index of graphs}. MATCH Communications in Mathematical and in Computer Chemistry 201879, 371--386, (2018).
\bibitem{Ramkumar} Ramkumar, R. and Nagarajan, K., {\it Greatest common divisor degree energy of graphs}. International Journal of Mathematical Sciences and Engineering Applications 11(II), 163--171, (2017).
\bibitem{Ramkumar} Ramkumar, R. and Nagarajan, K., {\it Greatest common divisor degree energy of graphs}. International Journal of Mathematical Sciences and Engineering Applications 11(II), 163--171, (2017).
\bibitem{Rilwan} Rilwan, N. M. and Hussain, A. S., {\it Different kinds of cordial labeling on identity graphs}. European Chemical Bulletin 12(10), 12374--12381, (2023).
\bibitem{Romdhini1} Romdhini, M. U. and Nawawi, A. and Syechah, B. N., {\it Seidel Laplacian and Seidel signless Laplacian energies of commuting graph for dihedral groups}. Malaysian Journal of Fundamental and Applied Sciences 20(3), 701--713, (2024).
\bibitem{Romdhini2} Romdhini, M. U. and Nawawi, A. and Al-Sharqi, F. and Al-Quran, A., {\it Characteristic polynomial of power graph for dihedral groups using degree-based matrices}. Malaysian Journal of Fundamental and Applied Sciences 20(2), 328--335, (2024).
\bibitem{Romdhini3} Romdhini, M. U. and Nawawi, A., {\it Maximum and minimum degree energy of commuting graph for dihedral groups}. Sains Malaysiana 51(12), 4145--4151, (2022).
\bibitem{Romdhini7} Romdhini, M. U. and Nawawi, A. and Al-Sharqi, F. and Salwa, {\it On Spectrum and Energy of Identity Graph for Group of Integers Modulo $n$, $\mathbb{Z}_n$}. European Journal of Pure and Applied Mathematics 17(4), 2915--2929, (2024).
Publicado
2026-01-22
Sección
Advances in Algebra, Analysis, Optimization, and Modeling
Derechos de autor 2026 Boletim da Sociedade Paranaense de Matemática
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