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Feynman graph polynomials. (English) Zbl 1193.81072

Summary: The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson’s identity and matroids.

MSC:

81T18 Feynman diagrams
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory

References:

[1] DOI: 10.1007/s00220-006-0040-2 · Zbl 1109.81059 · doi:10.1007/s00220-006-0040-2
[2] DOI: 10.4310/CNTP.2008.v2.n4.a1 · Zbl 1214.81095 · doi:10.4310/CNTP.2008.v2.n4.a1
[3] DOI: 10.1007/s11537-007-0648-9 · Zbl 1161.11021 · doi:10.1007/s11537-007-0648-9
[4] DOI: 10.1007/s00220-009-0740-5 · Zbl 1196.81130 · doi:10.1007/s00220-009-0740-5
[5] DOI: 10.1088/1751-8113/41/31/315402 · Zbl 1156.81036 · doi:10.1088/1751-8113/41/31/315402
[6] DOI: 10.4310/CNTP.2009.v3.n1.a1 · Zbl 1171.81384 · doi:10.4310/CNTP.2009.v3.n1.a1
[7] Marcolli M., Feynman Motives (2010)
[8] DOI: 10.1016/S0370-2693(02)02910-6 · Zbl 1001.81063 · doi:10.1016/S0370-2693(02)02910-6
[9] DOI: 10.1016/j.nuclphysb.2004.10.044 · Zbl 1119.81356 · doi:10.1016/j.nuclphysb.2004.10.044
[10] DOI: 10.1142/S0217751X08042869 · Zbl 1165.81350 · doi:10.1142/S0217751X08042869
[11] DOI: 10.1016/j.cpc.2009.11.007 · Zbl 1221.11183 · doi:10.1016/j.cpc.2009.11.007
[12] DOI: 10.1155/S107379280313142X · Zbl 1067.11075 · doi:10.1155/S107379280313142X
[13] Belkale P., Duke Math. J. 116 pp 147–
[14] DOI: 10.1140/epjc/s2003-01389-7 · Zbl 1099.81534 · doi:10.1140/epjc/s2003-01389-7
[15] DOI: 10.1016/j.cpc.2007.11.012 · Zbl 1196.81010 · doi:10.1016/j.cpc.2007.11.012
[16] DOI: 10.1063/1.3106041 · Zbl 1214.81096 · doi:10.1063/1.3106041
[17] Kirchhoff G., Ann. Phys. Chem. 72 pp 497–
[18] Eden R. J., The Analytic S-Matrix (1966)
[19] Hwa R. C., Homology and Feynman Integrals (1966)
[20] Nakanishi N., Graph Theory and Feynman Integrals (1971)
[21] Todorov I. T., Analytic Properties of Feynman Diagrams in Quantum Field Theory (1971)
[22] Zavialov O. I., Renormalized Quantum Field Theory (1990)
[23] Smirnov V. A., Feynman Integral Calculus (2006)
[24] Itzykson C., Quantum Field Theory (1980)
[25] Weinzierl S., Fields Inst. Commun. 50 pp 345–
[26] Tutte W. T., Encyclopedia of Mathematics and Its Applications 21, in: Graph Theory (1984)
[27] DOI: 10.1007/BF01608530 · Zbl 0927.05087 · doi:10.1007/BF01608530
[28] DOI: 10.1137/0603033 · Zbl 0495.05018 · doi:10.1137/0603033
[29] Chen W. K., Applied Graph Theory, Graphs and Electrical Networks (1982)
[30] DOI: 10.1016/0012-365X(92)00059-Z · Zbl 0838.05080 · doi:10.1016/0012-365X(92)00059-Z
[31] DOI: 10.1007/978-1-4613-0163-9 · doi:10.1007/978-1-4613-0163-9
[32] Sokal A. D., Surveys in Combinatorics (2005)
[33] DOI: 10.1017/S0305004100023173 · doi:10.1017/S0305004100023173
[34] DOI: 10.4153/CJM-1954-010-9 · Zbl 0055.17101 · doi:10.4153/CJM-1954-010-9
[35] DOI: 10.1016/S0021-9800(67)80032-2 · Zbl 0147.42902 · doi:10.1016/S0021-9800(67)80032-2
[36] Ellis-Monaghan J., Structural Analysis of Complex Networks (2010)
[37] Ellis-Monaghan J., Structural Analysis of Complex Networks (2010)
[38] DOI: 10.1098/rspl.1866.0037 · doi:10.1098/rspl.1866.0037
[39] Zeilberger D., Electron. J. Combin. 4 pp 2–
[40] DOI: 10.1007/BF01608531 · Zbl 0927.05002 · doi:10.1007/BF01608531
[41] Kuratowski C., Fund. Math. 15 pp 271–
[42] DOI: 10.1007/BF01594196 · Zbl 0017.19005 · doi:10.1007/BF01594196
[43] Diestel R., Graph Theory (2005)
[44] Oxley J., Matroid Theory (2006)
[45] Oxley J., Cubo 5 pp 179–
[46] DOI: 10.2307/2371127 · Zbl 0006.37005 · doi:10.2307/2371127
[47] Oxley J., Theory of Matroids (1986)
[48] DOI: 10.1002/jgt.3190040106 · Zbl 0397.05043 · doi:10.1002/jgt.3190040106
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