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- Title
Cycle/Cocycle Oblique Projections on Oriented Graphs.
- Authors
Polettini, Matteo
- Abstract
It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges to an arbitrarily chosen spanning tree. In this paper, we show that the edge vector space can also be decomposed in terms of cycles and the generating edges of cocycles (called cochords), or of cocycles and the generating edges of cycles (called chords). From this observation follows a construction in terms of oblique complementary projection operators. We employ this algebraic construction to prove several properties of unweighted Kirchhoff-Symanzik matrices, encoding the mutual superposition between cycles and cocycles. In particular, we prove that dual matrices of planar graphs have the same spectrum (up to multiplicities). We briefly comment on how this construction provides a refined formalization of Kirchhoff's mesh analysis of electrical circuits, which has lately been applied to generic thermodynamic networks.
- Subjects
OBLIQUE projection; SPANNING trees; DIRECTED graphs; MATHEMATICAL decomposition; MATHEMATICAL models; MATHEMATICAL functions
- Publication
Letters in Mathematical Physics, 2015, Vol 105, Issue 1, p89
- ISSN
0377-9017
- Publication type
Article
- DOI
10.1007/s11005-014-0732-z