Euler Characteristic of Graphs and Networks

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

commuting and non -commuting graphs of finit groups

فرض کنیمg یک گروه غیر آبلی متناهی باشد . گراف جابجایی g که با نماد نمایش داده می شود ،گرافی است ساده با مجموعه رئوس که در آن دو راس با یک یال به هم وصل می شوند اگر و تنها اگر . مکمل گراف جابجایی g راگراف نا جابجایی g می نامیم.و با نماد نشان می دهیم. گرافهای جابجایی و ناجابجایی یک گروه متناهی ،اولین بار توسطاردوش1 مطرح گردید ،ولی در سالهای اخیر به طور مفصل در مورد بحث و بررسی قرار گرفتند . در ،م...

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...