Hearing Euler characteristic of graphs

On the Dimension and Euler characteristic of random graphs

The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the distribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n− 1)/2 edges ap...

The generating polynomial and Euler characteristic of intersection graphs

Let E” be n-dimensional Euclidean space. A molecular space is a family of unit cubes in E”. Any molecular space can be represented by its intersection graph. Conversely, it is known that any graph G can be represented by molecular space M(G) in E” for some n. Suppose that S, and S, are topologically equivalent surfaces in E” and molecular spaces M, and M, are the two families of unit cubes inte...

Exponentiation and Euler Characteristic

The Euler Characteristic

I will describe a few basic properties of the Euler characteristic and then I use them to prove special case of a cute formula due to Bernstein-Khovanskii-Koushnirenko. 1. Basic properties of the Euler characteristic The Euler characteristic is a function χ which associates to each reasonable topological space X an integer χ(X). For us a reasonable space would be a space which admits a finite s...

The Euler Characteristic of an Enriched Category

We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category V is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler characteristic of V-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structu...