Let G denote a bipartite distance-regular graph with diameter D \ge 4 and valency k \ge 3. Given pseudo primitive idempotents E, F of G, we define the pair E, F to be taut whenever the entry-wise product of E and F is not a scalar multiple of a pseudo primitive idempotent, but is a linear combination of two pseudo primitive idempotents of G. In this paper, we determine all the taut pairs of pseudo primitive idempotents of G.

COBISS.SI-ID: 1541455556

Let G denote a distance-biregular graph with vertex set X. Fix vertex x in X and let T = T(x) denote the Terwilliger algebra of G with respect to x. In this paper we consider irreducible T-modules with endpoint 1. We show that there are no such modules if and only if G is the complete bipartite graph K_{1,n} and x is a vertex of G with valency 1. If the valency of x is at least 2, then we show that up to isomorphism there is a unique irreducible T-module of endpoint 1, and this module is thin.

COBISS.SI-ID: 1542038980

We introduce an infinite family of permutation groups, which are the complete automorphism groups of two different families of directed strongly regular graphs. For both families, there is a cyclic subgroup of the permutation group which acts semiregularly on the set of vertices of the directed graph and has two orbits. One of the two series gives an infinite number of directed strongly regular graphs admitting a cyclic semiregular automorphism group with structure and an automorphism group for which only three sporadic examples were previously known.

COBISS.SI-ID: 1541402308