In [S. Arumugam, V. Mathew and J. Shen, On fractional metric dimension of graphs, preprint], Arumugam et al. studied the fractional metric dimension of the cartesian product of two graphs, and proposed four open problems. In this paper, we determine the fractional metric dimension of vertex-transitive graphs, in particular, the fractional metric dimension of a vertex-transitive distance-regular graph is expressed in terms of its intersection numbers. As an application, we calculate the fractional metric dimension of Hamming graphs and Johnson graphs, respectively. Moreover, we give an inequality for metric dimension and fractional metric dimension of an arbitrary graph, and determine all graphs when the equality holds. Finally, we establish bounds on the fractional metric dimension of the cartesian product of graphs. As a result, we completely solve the four open problems.

Suzuki (2004) [7] classified thin weakly distance-regular digraphs and proposed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) [9] under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.

Let V be an n-dimensional vector space over a finite field F q . In this paper we describe the structure of maximal non-trivial t-intersecting families of k-dimensional subspaces of V with large size. We also determine the non-trivial t-intersecting families with maximum size. In the special case when t = 1 our result gives rise to the well-known Hilton-Milner Theorem for vector spaces.

A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify such digraphs with valency more than 3. As a result, this family of digraphs are completely determined.

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In [The Terwilliger algebra of the Johnson schemes, Discrete Mathematics 307 (2007) 1621-1635], Levstein and Maldonado computed the Terwilliger algebra of the Johnson scheme J(n, m) when 3m ≤ n. The distance-m graph of J(2m + 1, m) is the Odd graph Om+1. In this paper, we determine the Terwilliger algebra of Om+1 and give its basis.

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