%0 Journal Article %1 Wu2006138 %A Wu, Yaokun %A Deng, Aiping %D 2006 %J Linear Algebra and its Applications %K graph.theory hoffman linear.algebra matrix polynomial spectral_graph_theory %N 1 %P 138 -- 171 %R 10.1016/j.laa.2005.09.012 %T Hoffman Polynomials of Nonnegative Irreducible Matrices and Strongly Connected Digraphs %V 414 %X For a nonnegative n × n matrix A, we find that there is a polynomial f ( x ) ∈ R x such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.

@article{Wu2006138, abstract = {For a nonnegative n × n matrix A, we find that there is a polynomial f ( x ) ∈ R [ x ] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials. }, added-at = {2015-08-12T04:16:14.000+0200}, author = {Wu, Yaokun and Deng, Aiping}, biburl = {https://www.bibsonomy.org/bibtex/2c5e20404f0a0937cc49786854ae45c76/ytyoun}, doi = {10.1016/j.laa.2005.09.012}, interhash = {c0888050a8c61a15b84623d81af5fe06}, intrahash = {c5e20404f0a0937cc49786854ae45c76}, issn = {0024-3795}, journal = {Linear Algebra and its Applications }, keywords = {graph.theory hoffman linear.algebra matrix polynomial spectral_graph_theory}, number = 1, pages = {138 -- 171}, timestamp = {2015-11-06T07:52:56.000+0100}, title = {{Hoffman} Polynomials of Nonnegative Irreducible Matrices and Strongly Connected Digraphs }, volume = 414, year = 2006 }