**International Day of Mathematics**

**Aida Abiad**

#### What does the spectrum reveal about a graph?

In this talk we will look at the spectrum (eigenvalues) of the adjacency matrix of a graph, and ask whether the eigenvalues determine the graph. This is a difficult, but important problem which plays a special role in the famous graph isomorphism problem. It has been conjectured by van Dam and Haemers that almost every graph is determined by its spectrum. The mentioned problem has been solved for several families of graphs; sometimes by proving that the spectrum determines the graph, and sometimes by constructing nonisomorphic graphs with the same spectrum. In recent years this problem has attracted much interest. We will report on recent results concerning this conjecture.

**Pieter Kleer**

#### Generation of random graphs with given degrees

The problem of efficiently generating a graph with a given degree sequence is an important open problem. The goal is to come up with a randomized algorithm that outputs every graph (with the given degree sequence) with approximately the same probability. Such algorithms are used, e.g., for hypothesis testing in large (social) networks, and in the field of ecology. In this talk I will survey some of my work on this problem, and on variations in which additional constraints are imposed on the graphs that have to be generated.

**Diletta Martinelli**

#### Some thoughts about my mathematics experiences in Africa

In the past few years I had the opportunity to interact with several mathematical communities in Africa, in particular the African Institutes for Mathematical Studies in South Africa and Cameroon and the Mathematics Department at the University of Nairobi. I will share some thoughts and reflections about these experiences.