Georgia Institute of Technology College of Sciences

A well known result in graph theory states that a graph is connected if and only if the second eigenvalue of its Laplacian matrix is positive. In fact, the larger the second eigenvalue, the more connected the graph is. By varying the weights on edges, one can in general increase the second eigenvalue which in turn affects many graph properties such as expansion, mixing times of random walks etc.

In this talk, I will introduce conformally rigid graphs, which are those unweighted undirected graphs in which one cannot increase the second eigenvalue or decrease the largest eigenvalue by changing
weights. This notion turns out to be deeply connected to graph embeddings, semidefinite programming and other ideas in geometry, optimization and combinatorics.

Joint work with Joao Gouveia and Stefan Steinerberger