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Weighted Digraphs and its Spectrum

Author(s):Jimly Manuel, Aneesh Kumar K, Bijumon R

Affiliation: Department of Mathematics, Mahatma Gandhi College, Iritty, Department of Statistics, Mahatma Gandhi College, Iritty

Page No: 9-13

Volume issue & Publishing Year: Volume 2, Issue 3, March 2025

published on: 2025/03/30

Journal: International Journal of Advanced Multidisciplinary Application.(IJAMA)

ISSN NO: 3048-9350

DOI: https://doi.org/10.5281/zenodo.17331052

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Abstract:
This paper explores the concept of Weighted Digraphs associated with cyclic groups Zn. In these digraphs, each arc is assigned a weight based on modular arithmetic, specifically the smallest integer r such that y ? r x (mod n), where x and y are vertices of the digraph. The study discusses various properties of these weighted digraphs, including the relationship between the weight of arcs and the order of elements in Zn, the behavior of generators, and the uniqueness of arc weights. The adjacency matrix and degree matrix are defined, and the Laplacian matrix is derived as the difference between these two matrices. Several examples are presented for Z2, Z3, Z4, Z5, and Z6, showcasing their adjacency matrices, degree matrices, Laplacian matrices, characteristic polynomials, and eigenvalues. The results provide insight into the structure and properties of weighted digraphs on cyclic groups and demonstrate the use of software tools such as MATLAB for matrix computations

Keywords: Weighted digraph, Adjacency matrix, Spectrum, Laplacian matrix. 2000 Mathematics Subject Classification: 05C20, 05C50.

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