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.
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Ashtekar, A., Schilling, T.A.: Geometrical formulation of quantum mechanics. In: On Einsteins Path, pp. 23–65. Springer, New York (1999)
Bernstein, D.: Matrix Mathematics, vol. 44. Princeton University Press, Princeton (2005)
Biggs N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)
Bogner C., Weinzierl S.: Feynman graph polynomials. Int. J. Mod Phys A 25, 2585–2618 (2010)
Conder M., Robertson E., Williams P.: Presentations for 3-dimensional special linear groups over integer rings. Proc. Amer. Math. Soc. 115, 19–26 (1992)
Hill T.L.: Free Energy Transduction and Biochemical Cycle Kinetics. Dover, New York (2005)
Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)
Iyer T.S.K.V.: Circuit Theory. Tata Mac Graw-Hill, New Dehli (2006)
Lewkowicz I.: Bounds for the singular values of a matrix with nonnegative eigenvalues. Linear Algebra Appl. 112, 29 (1989)
Marcolli M.: Feynman Motives. World Scientific, Singapore (2009)
Nakanishi N.: Graph Theory and Feynman Integrals. Gordon and Breach, New York (1971)
Oster G., Perelson A., Katchalsky A.: Network thermodynamics. Nature 234, 393 (1971)
Polettini M.: Nonequilibrium thermodynamics as a Gauge theory. Europhys. Lett. 97, 30003 (2012)
Polettini M., Esposito M.: Irreversible thermodynamics of open chemical networks I: emergent cycles and broken conservation laws. J. Chem. Phys. 141, 024117 (2014)
Polettini, M., Esposito, M.: Transient fluctuation theorem for the currents and initial equilibrium ensembles. J. Stat. Mech. P10033 (2014)
Schnakenberg J.: Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571 (1976)
Szyld D.B.: The many proofs of an identity on the norm of oblique projections. Numer. Algor. 42, 309–323 (2006)
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Polettini, M. Cycle/Cocycle Oblique Projections on Oriented Graphs. Lett Math Phys 105, 89–107 (2015). https://doi.org/10.1007/s11005-014-0732-z
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DOI: https://doi.org/10.1007/s11005-014-0732-z