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  • eigenvalue  (1)
  • hyperplane arrangement  (1)
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  • 1
    Electronic Resource
    Electronic Resource
    Spectra of Some Interesting Combinatorial Matrices Related to Oriented Spanning Trees on a Directed Graph (1996)
    Springer
    Journal of algebraic combinatorics 5 (1996), S. 5-11 
    Details
    ISSN: 1572-9192
    Keywords: oriented spanning tree ; l-walk ; eigenvalue
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Laplacian of a directed graph G is the matrix L(G) = O(G) − A(G), where A(G) is the adjacency matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) and their multiplicities in terms of the eigenvalues of the induced subgraphs and the Laplacian matrix of G. Finally we compute the eigenvalues of D(G) for some specific directed graphs G. A recent conjecture of Propp for D(H n) follows, where H n stands for the complete directed graph on n vertices without loops.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022432212605
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley (1999)
    Springer
    Journal of algebraic combinatorics 10 (1999), S. 207-225 
    Details
    ISSN: 1572-9192
    Keywords: hyperplane arrangement ; characteristic polynomial ; root system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A hyperplane arrangement is said to satisfy the “Riemann hypothesis” if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system A n − 1. The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018778031648
    Library Location Call Number Volume/Issue/Year Availability
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    Paper (German National Licenses)
    Fulltext
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