Laplace and Dirac operators on graphs. (English) Zbl 1533.05151
Summary: Discrete versions of the Laplace and Dirac operators have been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper, we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equations. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 05C90 | Applications of graph theory |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 15A67 | Applications of Clifford algebras to physics, etc. |
| 35J10 | Schrödinger operator, Schrödinger equation |
Keywords:
graph Laplacian; Dirac operator; Schrödinger equation; Dirac equation; dimer model; Clifford algebra; graph gluingReferences:
| [1] | Cimasoni, D, Reshetikhin, N.Dimers on surface graphs and spin structures. i. Commun Math Phys. 2007;275(1):187-208. DOI: . · Zbl 1135.82006 |
| [2] | Mnev, P. Graph quantum mechanics. Contribution to the 2016 MPIM Jahrbuch. 2016. |
| [3] | Reshetikhin, N, Vertman, B.Combinatorial quantum field theory and gluing formula for determinants. Lett Math Phys. 2015;105(3):309-340. · Zbl 1312.58011 |
| [4] | Budinich, M, Budinich, P.A spinorial formulation of the maximum clique problem of a graph. J Math Phys. 2006;47(4):043502. · Zbl 1111.05073 |
| [5] | Staples, GS.Zeon matrix inverses and the zeon combinatorial Laplacian. Adv Appl Clifford Algebras. 2021;31(3):40. DOI: . · Zbl 1471.15002 |
| [6] | Harris, GM, Staples, GS.Spinorial formulations of graph problems. Adv Appl Clifford Algebras. 2012;22(1):59-77. DOI: . · Zbl 1266.15036 |
| [7] | Knill, O. The Dirac operator of a graph. 2013. Available at arXiv:math/1306.2166. · Zbl 1296.05136 |
| [8] | Kenyon, R.The Laplacian and Dirac operators on critical planar graphs. Invent Math. 2002;150(2):409-439. DOI: . · Zbl 1038.58037 |
| [9] | Khovanova, T. Clifford algebras and graphs. 2008. Available at arXiv:math/0810.3322. · Zbl 1506.15025 |
| [10] | Nica, B.A brief introduction to spectral graph theory. Zürich: European Mathematical Society; 2018. Available at https://www.amazon.com/Introduction-Spectral-Theory-Textbooks · Zbl 1403.05002 |
| [11] | Contreras, I, Xu, B.The graph Laplacian and morse inequalities. Pac J Math. 2019;300(2):331-345. DOI: . · Zbl 1434.05091 |
| [12] | Yu, C.Super-walk formulae for even and odd Laplacians in finite graphs. Rose-Hulman Undergrad Math J. 2017;18. Available at https://scholar.rose-hulman.edu/rhumj/vol18/iss1/16. |
| [13] | Contreras, I, Toriyama, M, Yu, C.Gluing of graph Laplacians and their spectra. Linear Multilinear Algebra. 2020;68(4):710-749. DOI: . · Zbl 1434.05090 |
| [14] | Goncharov, AB, Kenyon, R.Dimers and cluster integrable systems. Ann Sci Ec Norm Super. 2013;46(5):747-813. · Zbl 1288.37025 |
| [15] | Kasteleyn, P.Graph theory and crystal physics. In Harary, F., ed., Graph theory and theoretical physics. London: Academic Press; 1967. p. 43-110. · Zbl 0205.28402 |
| [16] | Renaud, P.Clifford algebras lecture notes on applications in physics. 2020. Available at https://hal.archives-ouvertes.fr/hal-03015551. |
| [17] | Haag, R, Łopuszański, JT, Sohnius, M.All possible generators of supersymmetries of the s-matrix. Nucl Phys B. 1975;88(2):257-274. Available at https://www.sciencedirect.com/science/article/pii/05503213759 |
| [18] | Iga, K.Adinkras graphs of Clifford algebra representations, supersymmetry, and codes. Adv Appl Clifford Algebras. 2021;31(5):76. DOI: . · Zbl 1478.15036 |
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