Related papers: The dimensions and Euler characteristics of M. Kon…
We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and…
We consider the problem of graph generation guided by network statistics, i.e., the generation of graphs which have given values of various numerical measures that characterize networks, such as the clustering coefficient and the number of…
A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$.…
Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the…
The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect…
Generalizing the star graphs of M\"uller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the…
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated…
We will consider P-graph complexes, where P is a cyclic operad. P-graph complexes are natural generalizations of Kontsevich's graph complexes -- for P = the operad for associative algebras it is the complex of ribbon graphs, for P = the…
A closed form formula (generating function) for the Euler characteristic of the configuration space of $\scriptstyle n$ particles in a simplicial complex is given.
A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…
Discovering the underlying structures present in large real world graphs is a fundamental scientific problem. In this paper we show that a graph's clique tree can be used to extract a hyperedge replacement grammar. If we store an ordering…
Let $V$ be a closed subscheme of a projective space $\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $ V$. The algorithm can be implemented using either symbolic or…
Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…
Substantial efforts have been made to compute or estimate the minimum number $c(G)$ of cycles needed to partition the edges of an Eulerian graph. We give an equivalent characterization of Eulerian graphs of treewidth $2$ and with maximum…
Random hypergraphs extend the classical notion of random graphs by allowing hyperedges to join more than two vertices, making them well-suited for modeling higher-order interactions in complex systems. Despite their broad applicability,…
In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves.
We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the…
In this paper we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex which is generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology…