Group A: The Graph Isomorphism Problem
The Graph Isomorphism Problem
Introduction
The graph isomorphism problem has been a long-standing problem in complexity theory for the last four decades. The statement of the problem is fairly simple: Given any two finite graphs that may appear to be different, is there an "easy" way to tell whether or not the two graphs are the same? We'll look at the statement of the problem in a slightly more rigorous setting, discuss the importance of the problem in other fields, as well as some recent progress made on the problem by a professor at the University of Chicago.
Basic concepts
We'll start off with some basic concepts related to the problem and formulate a more rigorous statement of the problem.
$ \begin{mydef} \text{Taylor Series in } d \text{ variables } =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\! \end{mydef} $
Importance
Recent developments
Closing Remarks
