Related papers: Dual Graph Polynomials and a 4-face Formula
The $c_2$ invariants in all 4 different representations of the Feynman period (parametric and dual parametric representations, position and momentum spaces) coincide for all log-divergent graphs that satisfy the combinatorial condition…
A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are…
The c_2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c_2 invariant in momentum space and prove that it equals the c_2 invariant in…
The obstruction for application of effective techniques like denominator reduction for the computation of the $c_2$ invariant of Feynman graphs in general is the absence of a 3-valent vertex for the initial steps. In this paper such a…
Feynman diagrams in $\phi^4$ theory have as their underlying structure 4-regular graphs. In particular, any 4-point $\phi^4$ graph can be uniquely derived from a 4-regular graph by deleting a vertex. The Feynman period is a simplified…
The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees…
A 4-point Feynman diagram in scalar $\phi^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is…
Double triangle expansion is an operation on $4$-regular graphs with at least one triangle which replaces a triangle with two triangles in a particular way. We study the class of graphs which can be obtained by repeated double triangle…
For every regular graph, we define a sequence of integers, using the recursion of the Martin polynomial. This sequence counts spanning tree partitions and constitutes the diagonal coefficients of powers of the Kirchhoff polynomial. We prove…
The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion…
We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a…
The $c_2$ invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$ invariant has a particular symmetry known as completion invariance. This…
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop…
We review an approach for the computation of Feynman integrals by use of multiple polylogarithms, with an emphasis on the related criterion of linear reducibility of the graph. We show that the set of graphs which satisfies the linear…
We introduce a new cohomology theory for planar trivalent graphs with perfect matchings. The graded Euler characteristic of the cohomology is a one variable polynomial called the 2-factor polynomial that, if nonzero when evaluated at one,…
We propose a framework for calculating two-loop Feynman diagrams which appear within a renormalizable theory in the general mass case and at finite external momenta. Our approach is a combination of analytical results and of high accuracy…
Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal…
We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal…
Feynman diagrams are a pictorial way of describing integrals predicting possible outcomes of interactions of subatomic particles in the context of quantum field physics. It is highly desirable to have an intrinsic mathematical…
We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph.…