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On the eccentricity matrices of line graphs of trees and wheel graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 10 مهر 1405 اصل مقاله (461.16 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31683.2927 | ||
| نویسنده | ||
| Shivani Chauhan* | ||
| Department of Applied Sciences and Humanities, ABES Engineering College, Ghaziabad, India-201009 | ||
| چکیده | ||
| The eccentricity matrix $\mathcal{E}(X)$ of a connected graph $X$ is obtained from the distance matrix of $X$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. In this paper, we investigate the spectral properties of the eccentricity matrices of line graphs of trees and wheel graphs. We prove that if $X$ is a tree on $n$ vertices of odd diameter $d \geq 5$, then the inertia of $\mathcal{E}(L(X))$ is $(2, 2, n-5)$. Furthermore, for trees of even diameter $d\geq 4$, we show that the inertia is $(l, l, n-2l-1)$, where $l$ denotes the number of diametrically distinguished vertices of $X$. Finally, we extend our analysis to determine the inertia, eigenvalues, and determinant of the eccentricity matrix for the line graphs of wheel graphs. We also relate its spectrum to the Dirichlet kernel. | ||
| کلیدواژه‌ها | ||
| Tree؛ line graph؛ eccentricity matrix | ||
| مراجع | ||
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[1] S. Balamoorthy, S.V. Bharanedhar, Y. Shang, and R. Gurusamy, On the eccentric spectra of the line graph of starlike trees, Discrete Dynamics in Nature and Society 2025 (2025), no. 1, 9972973. https://doi.org/10.1155/ddns/9972973
[2] L.W. Beineke and J.S. Bagga, Line Graphs and Line Digraphs, Springer, 2021.
[3] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012.
[4] P.J. Davis, Circulant Matrices, American Mathematical Soc., 2013.
[5] T. Divyadevi and I. Jeyaraman, On the eccentricity matrices of Certain bi-block graphs, Bull. Malays. Math. Sci. Soc. 47 (2024), no. 3, 92. https://doi.org/10.1007/s40840-024-01687-3
[6] F. Harary, Graph Theory, Addison-Wesley, 1969.
[7] E.V. Haynsworth, Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra Appl. 1 (1968), no. 1, 73–81. https://doi.org/10.1016/0024-3795(68)90050-5
[8] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 2012.
[9] I. Jeyaraman and T. Divyadevi, On eccentricity matrices of wheel graphs, Indian J. Pure Appl. Math. 56 (2025), no. 4, 1507–1522. https://doi.org/10.1007/s13226-024-00717-x
[10] X. Lei, and J. Wang, Spectral determination of graphs with one positive antiadjacency eigenvalue, Appl. Math. Comput. 422 (2022), 126995. https://doi.org/10.1016/j.amc.2022.126995
[11] X.H. Li, J.F. Wang, and M. Brunetti, Inertia and spectral symmetry for the eccentricity matrices of clique trees, Acta Math. Appl. Sin. Engl. Ser. 42 (2026), no. 1, 23–38. https://doi.org/10.1007/s10255-024-1140-5
[12] I. Mahato and M.R. Kannan, On the eccentricity matrices of trees: Inertia and spectral symmetry, Discrete Math. 345 (2022), no. 11, 113067. https://doi.org/10.1016/j.disc.2022.113067
[13] I. Mahato and M.R. Kannan, Extremal Problems for the Eccentricity Matrices of Complements of Trees, Electron. J. Linear Algebra 39 (2023), 339–354. https://doi.org/10.13001/ela.2023.7781
[14] S. Pandey, L. Selvaganesh, and J. Pervin, Eccentricity matrix of corona of two graphs, Discrete Appl. Math. 359 (2024), 354–363. https://doi.org/10.1016/j.dam.2024.08.017
[15] A.K. Patel, L. Selvaganesh, and S.K. Pandey, Energy and inertia of the eccentricity matrix of coalescence of graphs, Discrete Math. 344 (2021), no. 12, 112591. https://doi.org/10.1016/j.disc.2021.112591
[16] M. Randić, DMAX–matrix of dominant distances in a graph, MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 221–238.
[17] J.R. Silvester, Determinants of block matrices, Math. Gaz. 84 (2000), no. 501, 460–467.
[18] Y. Song, M. Brunetti, and J. Wang, On graphs with third largest eccentricity eigenvalue in the interval $[−2, −1]$, Discrete Appl. Math. 378 (2026), 671–681. https://doi.org/10.1016/j.dam.2025.09.027
[19] Y. Song, Y. Li, M. Brunetti, and J. Wang, On the third largest eigenvalue of eccentricity matrices of graphs, Discrete Appl. Math. 372 (2025), 237–259. https://doi.org/10.1016/j.dam.2025.04.039
[20] J. Wang, X. Lei, M. Lu, S. Sorgun, and H. Küçük, On graphs with exactly one anti-adjacency eigenvalue and beyond, Discrete Math. 346 (2023), no. 6, 113373. https://doi.org/10.1016/j.disc.2023.113373
[21] J. Wang, X. Lei, W. Wei, X. Luo, and S. Li, On the eccentricity matrix of graphs and its applications to the boiling point of hydrocarbons, Chemometrics and Intelligent Laboratory Systems 207 (2020), 104173. https://doi.org/10.1016/j.chemolab.2020.104173
[22] J. Wang, M. Lu, F. Belardo, and M. Randić, The anti-adjacency matrix of a graph: Eccentricity matrix, Discrete Appl. Math. 251 (2018), 299–309. https://doi.org/10.1016/j.dam.2018.05.062
[23] J. Wang, M. Lu, M. Brunetti, L. Lu, and X. Huang, Spectral determinations and eccentricity matrix of graphs, Adv. Appl. Math. 139 (2022), 102358. https://doi.org/10.1016/j.aam.2022.102358
[24] J. Wang, M. Lu, L. Lu, and F. Belardo, Spectral properties of the eccentricity matrix of graphs, Discrete Appl. Math. 279 (2020), 168–177. https://doi.org/10.1016/j.dam.2019.10.015
[25] J.F. Wang, X.Y. Lei, S.C. Li, and Z. Stani´c, The complete classification of graphs whose second largest eigenvalue of the eccentricity matrix is less than 1, Acta Math. Sin. (Engl. Ser.) 40 (2024), no. 7, 1741–1766. https://doi.org/10.1007/s10114-024-2413-x
[26] H. Whitney, Congruent graphs and the connectivity of graphs, Hassler Whitney Collected Papers, Springer, 1992, pp. 61–79. | ||
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