Let G denote a bipartite distance-regular graph with vertex set X and diameter D \ge 3. Fix x in X and let L (resp. R) denote the corresponding lowering (resp. raising) matrix. We show that each Q-polynomial structure for G yields a certain linear dependency among RL^2, LRL, L^2R, L. Define a partial order \le on X as follows. For y,z in X let y \le z whenever d(x,y)+d(y,z)=d(x,z), where d denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

COBISS.SI-ID: 1024466772

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence $\{3,4\}$ on atmost 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]} = 1$.

COBISS.SI-ID: 16699481

Let G denote a finite abelian group with identity 1 and let S denote an inverse-closed subset of G \ {1}, which generates G and for which there exists s in S, such that S \ {s,s^(-1)} does not generate the whole group G. In this paper we obtain the complete classification of distance-regular Cayley graphs Cay(G;S) for such pairs of G and S.