Graphs, ribbon graphs, and polynomials

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, different terminologies. We present here both terminologies, such that a sort of dictionary between these two communities can be established. We then extend the notion of graph to that of maps (or of ribbon graphs). Moreover, graph polynomials encoding these structures (the Tutte polynomial for graphs and the Bollobás–Riordan polynomial for ribbon graphs) are presented.

Quantum field theory (QFT)—built-in combinatorics

We briefly exhibit in this chapter the mathematical formalism of QFT, which actually has a non-trivial combinatorial backbone. The QFT setting can be understood as a quantum description of particles and their interactions, a description which is also compatible with Einstein's theory of special relativity. Within the framework of elementary particle physics (or high-energy physics), QFT led to the Standard Model of Elementary Particle Physics, which is the physical theory tested with the best accuracy by collider experiments. Moreover, the QFT formalism successfully applies to statistical physics, condensed matter physics and so on. We show in this chapter how Feynman graphs appear through the so-called QFT perturbative expansion, how Feynman integrals are associated to Feynman graphs and how these integrals can be expressed via the help of graph polynomials, the Kirchhoff–Symanzik polynomials. Finally, we give a glimpse of renormalization, of the Dyson–Schwinger equation and of the use of the so-called intermediate field method. This chapter mainly focuses on the so-called Phi? QFT scalar model.

On the Degeneracy of the Orbit Polynomial and Related Graph Polynomials

The orbit polynomial is a new graph counting polynomial which is defined as OG(x)=∑i=1rx|Oi|, where O1, …, Or are all vertex orbits of the graph G. In this article, we investigate the structural properties of the automorphism group of a graph by using several novel counting polynomials. Besides, we explore the orbit polynomial of a graph operation. Indeed, we compare the degeneracy of the orbit polynomial with a new graph polynomial based on both eigenvalues of a graph and the size of orbits.

Graph polynomials and symmetries

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions and external fields are absolutely bounded close to zero. Furthermore, we prove that for this class of Ising models the partition function does not vanish. Our algorithm is based on an approach due to Barvinok for approximating evaluations of a polynomial based on the location of the complex zeros and a technique due to Patel and Regts for efficiently computing the leading coefficients of graph polynomials on bounded degree graphs. Finally, we show how our algorithm can be extended to approximate certain output probability amplitudes of quantum circuits.