آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 122 |
| تعداد مقالات | 1,538 |
| تعداد مشاهده مقاله | 1,628,301 |
| تعداد دریافت فایل اصل مقاله | 1,525,832 |
Spectral properties of a new distance-based matrix | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 14 اردیبهشت 1405 اصل مقاله (435.26 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31172.2767 | ||
| نویسندگان | ||
| Razie Alidehi-Ravandi؛ Modjtaba Ghorbani* | ||
| Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran | ||
| چکیده | ||
| In this paper, we introduce and analyze the spectral properties of the Graovac-Ghorbani matrix $\tilde{\mathcal{A}}$. We first calculate the Graovac-Ghorbani index for specific classes of structures, including nicely distance-balanced, vertex-transitive, and edge-transitive graphs. By defining the Laplacian matrix associated with $\tilde{\mathcal{A}}$, we prove that for any connected graph of order $n \ge 3$, the rank of this Laplacian matrix is exactly $n-1$. We establish sharp upper and lower bounds for the spectral radius of $\tilde{\mathcal{A}}$ in terms of graph parameters for various families, including trees, unicycle bipartite graphs, and general bipartite graphs, characterizing the extremal cases such as complete bipartite graphs. Furthermore, the spectral properties of strongly distance-balanced (SDB) graphs are investigated. A key finding of this study is the proof that the $\tilde{\mathcal{A}}$-eigenvalues of any bipartite graph are symmetric with respect to zero. However, we demonstrate that the converse does not hold, implying that a symmetric $\tilde{\mathcal{A}}$-spectrum does not necessarily guarantee the bipartiteness of the graph. | ||
| کلیدواژهها | ||
| Graovac-Ghorbani matrix؛ spectral radius؛ Rayleigh quot | ||
| مراجع | ||
|
[1] R. Alidehi-Ravandi and H.O. Abdullah, Topological indices of drug molecular structures: An application with the treatment and prevention of covid-19, J. Discrete Math. Appl. 8 (2023), no. 2, 81–101. https://doi.org/10.22061/jdma.2023.1944
[2] N. Amraei, A generalized version of symmetric division degree index, J. Discrete Math. Appl. 7 (2022), no. 3, 119–126. https://doi.org/10.22061/jdma.2023.2072 [3] R.B. Bapat, Graphs and Matrices, Springer, London, 2014.
[4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976.
[5] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer, New York, 2010.
[6] D. Cvetković and M. Doob, Developments in the theory of graph spectra, Linear Multilinear Algebra 18 (1985), no. 2, 153–181. https://doi.org/10.1080/03081088508817683
[7] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs–Theory and Application, Barth, Heidelberg, 1995.
[8] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs–Theory and Application, Deutscher Verlag der Wissenschaften, Berlin; Academic Press, New York, 1980.
[9] D. Dimitrov, B. Ikica, and R. krekovski, Remarks on the Graovac–Ghorbani index of bipartite graphs, Appl. Math. Comput. 293 (2017), 370–376. https://doi.org/10.1016/j.amc.2016.08.047
[10] S. Filipovski, Connected graphs with maximal Graovac–Ghorbani index, MATCH Commun. Math. Comput. Chem. 89 (2023), no. 2, 517–525. https://doi.org/10.46793/match.89-2.517F
[11] S. Furtula, Atom-bond connectivity index versus Graovac-Ghorbani analog, MATCH Commun. Math. Comput. Chem. 75 (2016), no. 1, 233–242.
[12] L. Gan, H. Hou, and B. Liu, Some results on atom-bond connectivity index of graphs, MATCH Commun. Math. Comput. Chem. 66 (2011), no. 2, 669–680.
[13] M. Ghorbani, Some GAP programs for computing the topological indices, J. Discrete Math. Appl. 7 (2022), no. 3, 147–154. https://doi.org/10.22061/jdma.2022.877
[14] M. Ghorbani and R. Alidehi-Ravandi, Exploring the SDE index: A novel approach using eccentricity in graph analysis, J. Appl. Math. Comput. 70 (2024), no. 2, 947–967. https://doi.org/10.1007/s12190-023-01980-7
[15] M. Ghorbani, R. Alidehi-Ravandi, N. Amraei, , and Z. Raza, On the largest eigenvalue of extended eccentric matrix, Chaos Solitons Fractals 185 (2024), 115074. https://doi.org/10.1016/j.chaos.2024.115074
[16] M. Ghorbani, Z. Vaziri, R. Alidehi-Ravandi, and Y. Shang, The symmetric division szeged index: A novel tool for predicting physical and chemical properties of complex networks, Heliyon 11 (2025), no. 3, e42280. https://doi.org/10.1016/j.heliyon.2025.e42280 [17] A. Graovac and M. Ghorbani, A new version of atom-bond connectivity index, Acta Chim. Slov 57 (2010), no. 3, 609–612.
[18] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N.Y. 27 (1994), no. 9, 9–15.
[19] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka 18 (2011), 17–23.
[20] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2012.
[21] K. Kutnar, A. Malnič, D. Marušič, and Š. Miklavič, The strongly distance-balanced property of the generalized petersen graphs, Ars Math. Contemp. 2 (2009), no. 1, 41–47. https://doi.org/10.26493/1855-3974.75.895
[22] S. Majstorović, Maximizing Graovac-Ghorbani index of trees with fixed maximum degree, MATCH Commun. Math. Comput. Chem. 90 (2023), no. 3, 673–684. https://doi.org/10.46793/match.90-3.673M
[23] S.C. Manju, K. Somasundaram, and Y. Shang, A new method for computing the vertex PI index with applications to special classes of graphs, AKCE Int. J. Graphs Comb. 22 (2025), no. 2, 673–684. https://doi.org/10.1080/09728600.2024.2424317
[24] B. Mohar, The Laplacian spectrum of graphs, Wiley, New York, 1991.
[25] I. Nadeem, S. Siddique, and Y. Shang, Some inequalities between general Randi´c-type graph invariants, J. Math. 2024 (2024), no. 1, 8204742. https://doi.org/10.1155/2024/8204742
[26] D. Pacheco, L. de Lima, and C.S. Oliveira, On the Graovac-Ghorbani index for bicyclic graphs with no pendent vertices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 2, 429–448.
[27] D. Pacheco, C.S. Oliveira, and A. Novanta, A survey on Graovac-Ghorbani index, MATCH Commun. Math. Comput. Chem. 90 (2023), no. 2, 301–312. https://doi.org/10.46793/match.90-2.301P
[28] B.A. Rather, H.A. Ganie, and Y Shang, On the signless Laplacian ABC-spectral properties of a graph, Mathematics 12 (2024), no. 15, 2366. https://doi.org/10.3390/math12152366
[29] M. Rostami and M. Sohrabi-Haghighat, Further results on new version of atom–bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 1, 21–32.
[30] R. Song, S. Liu, J. Ou, and J. Cui, On bicyclic graphs with maximal GraovacGhorbani index, MATCH Commun. Math. Comput. Chem. 91 (2024), no. 2, 513–532. https://doi.org/10.46793/match.91-2.513S | ||
|
آمار تعداد مشاهده مقاله: 20 تعداد دریافت فایل اصل مقاله: 24 |
||