Abstract: The eigenvalues of the adjacency matrix of a graph contain a lot --- but not always all --- information onthe structure of the graph. In this talk, we will dive deeper into graphs that have a lot of combinatorial symmetry:distance-regular graphs (such as Hamming graphs and Johnson graphs). We will give an overview of whendistance-regularity is determined by the eigenvalues (and when it is not). We will see how systems of orthogonalpolynomials can help to recognize distance-regular graphs from their eigenvalues and a little extra information through the `spectral excess theorem'.
We then discuss how these methods and ideas led to the construction of the twisted Grassmann graphs, a family ofdistance-regular graphs that have the same spectrum as certain Grassmann graphs. These twisted graphs are currently the only known family of distance-regular graphs with unbounded diameter that are not vertex-transitive.
If time permits, we also present some more recent results, such as a characterization of the generalized odd graphs ('the odd-girth theorem'), and discuss some results on graphs that are `almost distance-regular', in particular how the latter can be used to construct non-isomorphic graphs with the same eigenvalues.