On reversal of graph orientation
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V.A. Kolmykov
Abstract
Let each vertex of a finite directed graph be associated with a finite-dimensional linear space and each arc, with a linear transformation of the corresponding space. Such objects are referred to as linear representations of graphs. They naturally arise in some fields of algebra and are deeply studied in the past three decades.
Replacing all arcs entering into a sink by oppositely oriented ones, we arrive at a new directed graph. These two directed graphs are close to each other in the sense that the problem of classification of their representation are equivalent, as shown by Bernstein, Gelfand, and Ponomarev. Two orientations are equivalent if one is derived from another by means of a sequence of the above transformations.
In the directed graph representation theory, of most interest are circuit-free orientations. In this paper, we give a simple criterion for equivalence of circuit-free orientations. We prove that two orientations are equivalent if and only if some integrals of these orientations are equal to each other.
© 2016 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Editorial
- On permanents of random doubly stochastic matrices and asymptotic estimates of the nom hers of Latin rectangles and Latin squares
- On reversal of graph orientation
- The structure of an infinite Sylow subgroup in some periodic Shunkov groups
- Boolean lattices of multiply Ω-foliated formations
- Boolean lattices of n-multiply Ω-bicanonical Fitting classes
- On infinite groups with points
- Linear recurring sequence decimations with exponentially increasing step
- On ω-languages of special billiards
- On relative complexity of quantum and classical branching programs
- Forthcoming Papers
- Contents
Articles in the same Issue
- Editorial
- On permanents of random doubly stochastic matrices and asymptotic estimates of the nom hers of Latin rectangles and Latin squares
- On reversal of graph orientation
- The structure of an infinite Sylow subgroup in some periodic Shunkov groups
- Boolean lattices of multiply Ω-foliated formations
- Boolean lattices of n-multiply Ω-bicanonical Fitting classes
- On infinite groups with points
- Linear recurring sequence decimations with exponentially increasing step
- On ω-languages of special billiards
- On relative complexity of quantum and classical branching programs
- Forthcoming Papers
- Contents
