On eccentricity spectra and Wiener index of comaximal graph

Ali Yahya Hummdi; L. K. Abbiramy; K. Selvakumar; Junaid Nisar; Ali Yahya Hummdi; L. K. Abbiramy; K. Selvakumar; Junaid Nisar

doi:10.3934/math.20261246

AIMS Mathematics

2026, Volume 11, Issue 9: 31596-31609. doi: 10.3934/math.20261246

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On eccentricity spectra and Wiener index of comaximal graph

1.
Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia
2.
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India
3.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune 412115, India

Received: 01 July 2026 Revised: 13 September 2026 Accepted: 20 September 2026 Published: 24 September 2026
MSC : 05C12, 05C50

The eccentricity matrix $ \epsilon(G) $ of a connected graph $ G $ is a distance-based structural refinement constructed by preserving only the maximum geodesic distances within each row and column of the classical distance matrix, while mapping all remaining coefficients to zero. This paper investigated the architectural, spectral, and distance-based properties of the eccentricity matrix for comaximal graph $ \Gamma(R) $ associated with a general commutative Artinian ring $ R $ with unity is investigated. Extending this analysis to the ring of integers modulo $ n $, $ \mathbb{Z}_n $, we provided explicit determinations of the eccentricity characteristic polynomials, eccentricity spectra, and various eccentric connectivity parameters.
- distance matrix,
- eccentricity matrix,
- eigenvalue
Citation: Ali Yahya Hummdi, L. K. Abbiramy, K. Selvakumar, Junaid Nisar. On eccentricity spectra and Wiener index of comaximal graph[J]. AIMS Mathematics, 2026, 11(9): 31596-31609. doi: 10.3934/math.20261246

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Abstract

The eccentricity matrix $ \epsilon(G) $ of a connected graph $ G $ is a distance-based structural refinement constructed by preserving only the maximum geodesic distances within each row and column of the classical distance matrix, while mapping all remaining coefficients to zero. This paper investigated the architectural, spectral, and distance-based properties of the eccentricity matrix for comaximal graph $ \Gamma(R) $ associated with a general commutative Artinian ring $ R $ with unity is investigated. Extending this analysis to the ring of integers modulo $ n $, $ \mathbb{Z}_n $, we provided explicit determinations of the eccentricity characteristic polynomials, eccentricity spectra, and various eccentric connectivity parameters.

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