Weighted Digraphs and its Spectrum

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