PP-polynomial weakly distance-regular digraphs

A weakly distance-regular digraph is $P$-polynomial if its attached scheme is $P$-polynomial. In this paper, we characterize all $P$-polynomial weakly distance-regular digraphs

Finite ss-geodesic-transitive digraphs

This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $\Gamma$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $\Gamma_N$ is $(G/N,s')$-geodesic-transitive where s'=\min\{s,\diam(\Gamma_N)\}, $G/N$ is either quasiprimitive or bi-quasiprimitive on $V(\Gamma_N)$, and $\Gamma_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $\Gamma$ is not $(G,2)$-arc-transitive, then $G/N$ is quasiprimitive on $V(\Gamma_N)$. On the other hand, we also consider the case that the normal subgroup $N$ of $G$ has one orbit on the vertex set. We show that if $N$ is regular on $V(\Gamma)$, then $\Gamma$ is a circuit, and particularly each $(G,s)$-geodesic-transitive normal Cayley digraph with $s\geq 2$, is a circuit. Finally, we investigate $(G,2)$-geodesic-transitive digraphs with either valency at most 5 or diameter at most 2. Let $\Gamma$ be a $(G,2)$-geodesic-transitive digraph. It is proved that: if $\Gamma$ has valency at most $5$, then $\Gamma$ is $(G,2)$-arc-transitive; if $\Gamma$ has diameter $2$, then $\Gamma$ is a balanced incomplete block design with the Hadamard parameters

Weakly distance-regular circulants, I

We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.Comment: 28 page

A spectral excess theorem for digraphs with normal Laplacian matrices

The spectral excess theorem‎, ‎due to Fiol and Garriga in 1997‎, ‎is an important result‎, ‎because it gives a good characterization‎ ‎of distance-regularity in graphs‎. ‎Up to now‎, ‎some authors have given some variations of this theorem‎. ‎Motivated by this‎, ‎we give the corresponding result by using the Laplacian spectrum for digraphs‎. ‎We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues‎, ‎is distance-regular‎. ‎Hence such a digraph is strongly regular with girth $g=2$ or $g=3$‎