Homomorphisms, representations and characteristic polynomials of digraphs

AbstractThe existence of a homomorphism between two digraphs often implies many structural properties. We collect in this paper some characterizations of various digraph homomorphisms using matrix equations and fiber partitions. We also survey the relationship among the characteristic polynomials of a digraph and its divisors. This includes an introduction of the concept of branched coverings of digraphs, their voltage assignment representations, and a decomposition formula for the characteristic polynomial of a branched cover with branch index 1. Some open problems are included

Weighted zeta functions for quotients of regular coverings of graphs

AbstractLet G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs

Characteristic polynomials of ramified uniform covering digraphs

We give a decomposition formula for the characteristic polynomials of ramified uniform covers of digraphs. Similarly, we obtain a decomposition formula for the characteristic polynomials of ramified regular covers of digraphs. As applications, we establish decomposition formulas for the characteristic polynomials of branched covers of digraphs and the zeta functions of ramified covers of digraphs. Key words: characteristic polynomial, adjacency matrix, voltage digraph, ramified uniform cover, ramified regular cover, zeta function

Invariant subspace, determinant and characteristic polynomials ∗

Making use of an elementary fact on invariant subspace and determinant of a linear map and the method of algebraic identities, we obtain a factorization formula for a general characteristic polynomial of a matrix. This answers a question posed in [A. Deng, I. Sato, Y. Wu, Characteristic polynomials of ramified uniform covering digraphs, European Journal of Combinatorics 28 (2007), 1099–1114]. The approach of this work can be used to supply alternative proofs of several other earlier results, including some results of [Y. Teranishi, Equitable switching and spectra of graphs, Linear Algebra and its Applications 359 (2003), 121–131]