Homomorphisms, representations and characteristic polynomials of digraphs

The 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.

[1]  Jin Ho Kwak,et al.  Caracteristics polynomials of some grap bundlesII , 1992 .

[2]  A. Schwenk.  COMPUTING THE CHARACTERISTIC POLYNOMIAL OF A GRAPH. , 1974 .

[3]  Chris Godsil,et al.  Symmetry and eigenvectors , 1997 .

[4]  Sebastiano Vigna,et al.  Fibrations of graphs , 2002, Discret. Math..

[5]  Masakazu Nasu.  Constant-to-one and onto global maps of homomorphisms between strongly connected graphs , 1983 .

[6]  Yaokun Wu,et al.  Characteristic polynomials of digraphs having a semi-free action , 2005 .

[7]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[8]  Partitioned Tensor Products and Their Spectra , 1995, math/9503233.

[9]  P. Hr.  FRONT-DIVISORS OF TREES , 1992 .

[10]  M. Liebeck,et al.  Representations and Characters of Groups , 1995 .

[11]  Iwao Sato,et al.  Characteristic Polynomials of some covers of symmetric digraphs , 1997, Ars Comb..

[12]  Roman Nedela,et al.  FRONT-DIVISORS OF TREES , 1992 .

[13]  G. Sabidussi,et al.  Graph symmetry : algebraic methods and applications , 1997 .

[14]  Iwao Sato,et al.  Characteristic polynomials of ramified uniform covering digraphs , 2007, Eur. J. Comb..

[15]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[16]  A Note on a Theorem of Horst Sachs , 2005 .

[17]  Jonathan L. Gross,et al.  Generating all graph coverings by permutation voltage assignments , 1977, Discret. Math..

[18]  Brian Marcus,et al.  Topological entropy and equivalence of dynamical systems , 1979 .

[19]  Yaokun Wu,et al.  Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs , 2006 .

[20]  Edwin R. Hancock,et al.  Eigenspaces for Graphs , 2002, Int. J. Image Graph..

[21]  Iwao Sato,et al.  Characteristic polynomials of some graph coverings , 1995, Discret. Math..

[22]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[23]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[24]  Hye Kyung Kim,et al.  Characteristic polynomials of graphs having a semifree action , 2000 .

[25]  R. F. Williams.  Classification of subshifts of finite type , 1973 .

[26]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[27]  B. Kitchens.  Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts , 1997 .

[28]  Jin Ho Kwak,et al.  Characteristic polynomials of graph coverings , 2004 .