Homology, Homotopy and Applications

Volume 15 (2013)

Number 2

On Kirchhoff’s theorems with coefficients in a line bundle

Pages: 267 – 280

DOI: https://dx.doi.org/10.4310/HHA.2013.v15.n2.a16

Authors

Michael J. Catanzaro (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

Vladimir Y. Chernyak (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

John R. Klein (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

Abstract

We prove “twisted” versions of Kirchhoff’s network theorem and Kirchhoff’s matrix-tree theorem on connected finite graphs. Twisting here refers to chains with coefficients in a flat unitary line bundle.

Keywords

graph, homology, line bundle, Kirchhoff, matrix-tree

2010 Mathematics Subject Classification

Primary 57M15, 57Q10. Secondary 05C05, 05C21, 05E45, 82C31.

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Published 4 December 2014