Abstract:
A k-tree is a natural generalisation of a tree. It is any graph obtained by considering a (k+1)-clique and appending a new vertex to one of its k-cliques, then iterating this process on the resulting graph. The long standing open problem of the enumeration of unlabelled k-trees was solved by Gainer-Dewar in 2012, then later simplified by Gainer-Dewar and Gessel. The latter allowed for their asymptotic enumeration by Drmota and Yu Jin. In this talk we will discuss the enumeration of unlabelled chordal graphs with bounded tree-width, a generalisation of k-trees, via an extension of Pólya theory and Otter's dissimilarity method to graphs rooted a cliques. This is joint work with Jordi Castellví.