Feynman identity for planar graphs

TL;DR

This paper explores the Feynman identity for planar graphs, linking graph polynomials to free Lie superalgebras, and computes cycle class counts, with implications for graph zeta functions.

Contribution

It establishes the Feynman identity as a denominator identity for free Lie superalgebras generated by planar graphs, solving a problem posed by Sherman.

Findings

Computed the number of equivalence classes of nonperiodic cycles in planar graphs.

Interpreted the Feynman identity in the context of free Lie superalgebras.

Connected the results to zeta functions of graphs.

Abstract

The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by S. Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other results are obtained. For instance, in connection with zeta functions of graphs.