Cycle/cocycle oblique projections on oriented graphs

TL;DR

This paper introduces a novel decomposition of the edge vector space of oriented graphs using cycles and cocycles, employing oblique projections to analyze properties of Kirchhoff-Symanzik matrices and dual spectra in planar graphs.

Contribution

It presents a new decomposition method based on cycles and cochords or chords, and develops an algebraic framework with oblique projections for graph analysis.

Findings

Dual Kirchhoff-Symanzik matrices of planar graphs share the same spectrum.

The new decomposition refines Kirchhoff's mesh analysis for electrical circuits.

Properties of unweighted Kirchhoff-Symanzik matrices are proved using the algebraic construction.

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.