Algebraic invariants of weighted oriented graphs

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

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Counting Hamilton Decompositions of Oriented Graphs

International Mathematics Research Notices ◽

10.1093/imrn/rnx085 ◽

2017 ◽

Vol 2018 (22) ◽

pp. 6908-6933 ◽

Cited By ~ 1

Author(s):

Asaf Ferber ◽

Eoin Long ◽

Benny Sudakov

Keyword(s):

Oriented Graphs ◽

Hamilton Decompositions

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Towards the Characterization of Finite Homomorphism-Homogeneous Oriented Graphs with Loops

Graphs and Combinatorics ◽

10.1007/s00373-014-1435-z ◽

2014 ◽

Vol 31 (5) ◽

pp. 1613-1628 ◽

Cited By ~ 1

Author(s):

Dragan Mašulović

Keyword(s):

Oriented Graphs

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On the oriented chromatic index of oriented graphs

Journal of Graph Theory ◽

10.1002/jgt.20286 ◽

2008 ◽

Vol 57 (4) ◽

pp. 313-332 ◽

Cited By ~ 4

Author(s):

Pascal Ochem ◽

Alexandre Pinlou ◽

Éric Sopena

Keyword(s):

Chromatic Index ◽

Oriented Graphs

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Properties of Pseudo-Holomorphic Curves in Symplectisations II: Embedding Controls and Algebraic Invariants

Geometries in Interaction ◽

10.1007/978-3-0348-9102-8_6 ◽

1995 ◽

pp. 270-328

Author(s):

H. Hofer ◽

K. Wysocki ◽

E. Zehnder

Keyword(s):

Holomorphic Curves ◽

Algebraic Invariants

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Consecutive Colouring of Oriented Graphs

Results in Mathematics ◽

10.1007/s00025-021-01505-3 ◽

2021 ◽

Vol 76 (4) ◽

Author(s):

Marta Borowiecka-Olszewska ◽

Ewa Drgas-Burchardt ◽

Nahid Yelene Javier-Nol ◽

Rita Zuazua

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Bipartite Graphs ◽

Complete Graphs ◽

Oriented Graphs ◽

Complete Problem ◽

Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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