Unbalanced distance-biregular graphs with girth deficiency two

RESUMO

Let $\Gamma=(V,E)$ be a distance-biregular graph with even diameter $d$ and girth $2d-2$. On the edge-set $E$ we define relations that form an association scheme; this is done by considering the linear span of the corresponding adjacency matrices, and, by means of intersection diagrams, we prove that this is an algebra. Furthermore, we describe all the irreducible representations of this algebra, and, by considering the multiplicities of the 2-dimensional ones, we show that there are no such distance-biregular graphs with $28\le d\le 46$ and $d\ge 52$.