Related papers: The Euler characteristic of graph complexes via Fe…
In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to…
The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra.…
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular…
Kontsevich has proven that the Lie homology of the Lie algebra of symplectic vector fields can be computed in terms of the homology of a graph complex. We prove that the Leibniz homology of this Lie algebra can be computed in terms of the…
This paper explains how to use quantum field theory techniques to find formal power series that encode the virtual Euler characteristics of $\mathrm{Out}(F_n)$ and related graph complexes. Finding such power series was a necessary step in…
An Eulerian-minor of an Eulerian graph is obtained from an Eulerian subgraph of the Eulerian graph by contraction. The Eulerian-minor operation preserves Eulerian properties of graphs and moreover Eulerian graphs are well-quasi-ordered…
We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials…
For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links…
The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The…
We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used…
For a possibly singular complex variety $X$, generating functions of total "orbifold Chern homology classes" of symmetric products $S^nX$ are given. Those are very natural "Chern class versions" (in the sense of Schwartz-MacPherson) of…
Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that…
In this paper, we derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of…
We revisit and give a detailed proof of a lemma of Okounkov showing that, for a scheme X with a torus action, the Euler characteristic generating function associated with a "factorisable" sequence of torus-equivariant coherent sheaves on…
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated…
The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
We solve the problem of the computation of the orbifold Euler characteristics of $\Mbar_{g,n}$. We take the works of Harer-Zagier \cite{hz} and Bini-Harer \cite{bh} as our starting point, and apply the formalisms developed in \cite{wz} and…
In this paper, we derive eight basic identities of symmetry in three variables related to Euler polynomials and alternating power sums. These and most of their corollaries are new, since there have been results only about identities of…
We prove that a planar graph is Gorenstein if and only if its independence complex is Eulerian.