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Abstract: Let V be one of n-dimensional classical polar spaces over a finite field with q elements. Then all subspaces of V form a graded poset ordered by inclusion, denoted by . Given a fixed maximal totally isotropic subspace of V. Then each set is a graded subposet of , where . In this paper we show that has the NM property, which implies that has the strong Sperner property and the LYM property. [Copyright &y& Elsevier]

Abstract: As a generalization of attenuated spaces, the concept of singular linear spaces was introduced in [K. Wang, J. Guo, F. Li, Association schemes based on attenuated spaces, European J. Combin. 31 (2010) 297–305]. This paper first gives two anzahl theorems in singular linear spaces, and then discusses their applications to the constructions of Deza digraphs, quasi-strongly regular graphs, lattices and authentication codes. [Copyright &y& Elsevier]

Abstract: An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated. [Copyright &y& Elsevier]

Abstract: Let be the -dimensional symplectic space over a finite field . For and , let be a fixed subspace of type in , and let be the set of the 1-dimensional subspaces in not contained in . Denote by the graph with the vertex set , in which two vertices and are adjacent if is non-isotropic. Note that is a subgraph of the symplectic graph . In this paper, we determine an equitable partition of and compute its quotient matrix. Moreover, we show that is a strictly Deza graph, compute its parameters and spectra. Finally, we prove that the (fractional) chromatic number of is . [ABSTRACT FROM AUTHOR]

Abstract: The subspaces of a given dimension in an attenuated space form the points of an association scheme. If the dimension is maximal, it is the bilinear forms graph, which has been thoroughly studied. In this paper, we discuss the general case, and obtain a family of symmetric association schemes which are a common generalization of the Grassmann schemes and the bilinear forms schemes. Moreover, all the parameters of the association scheme are computed. [Copyright &y& Elsevier]

Abstract: Let be the n-dimensional vector space over a finite field , and let be the symplectic group where ; or the unitary group where . For any two orbits and of subspaces under , let (resp. ) be the set of all subspaces which are sums (resp. intersections) of subspaces in (resp. ) such that (resp. ). Suppose is the intersection of and containing {0} and . By ordering by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in these lattices and classifies their geometricity. [Copyright &y& Elsevier]

Abstract: Motivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Mathematics 282 (2004) 163–169] for a systematic way of constructing pooling designs; note that geometric lattices are among pooling spaces. This paper attempts to draw possible connections from finite geometry and distance regular graphs to pooling spaces: including the projective spaces, the affine spaces, the attenuated spaces, and a few families of geometric lattices associated with the orbits of subspaces under finite classical groups, and associated with -bounded distance-regular graphs. [Copyright &y& Elsevier]

Abstract: Let be one of n-dimensional affine-classical spaces over a finite field , and let be an orbit of totally isotropic flats under the corresponding affine-classical group. Denote by the set of all the intersections of totally isotropic flats in . If we partially order by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This paper discusses their geometricity and computes their characteristic polynomials. [Copyright &y& Elsevier]

Abstract: A necessary and sufficient condition is obtained for the lexicographic (resp. categorical) product from a Deza digraph to another to be also a Deza digraph; moreover, constructions of Deza digraphs from the incidence structures associated with vector spaces (resp. distance-regular graphs) are given. [Copyright &y& Elsevier]

Abstract: Let be a finite set associated with cliques of a distance-regular graph of order , with -cliques of Johnson graphs and antipodal distance-regular graphs of diameter , respectively. If we partially order by the ordinary inclusion, three families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials. [Copyright &y& Elsevier]

Abstract: A graph is half-transitive if its automorphism group acts transitively on its vertex set and edge set, but not on its arc set. In this paper, the tetravalent half-transitive graphs of order are classified for each prime . It is shown that there are no tetravalent half-transitive Cayley graphs of order and a tetravalent half-transitive non-Cayley graph of order exists if and only if is divisible by 8, which is unique for a given order. [Copyright &y& Elsevier]

Abstract: We consider a directed version of Deza graphs. A digraph is said to be a Deza digraph if it is regular and the number of common out-neighbors of any two distinct vertices takes on at most two values. We introduce some constructions and develop some basic theory. In particular, constructions from Cayley digraphs, graph lexicograph products and graph directed products are described. Moreover, we characterize a family of normal Deza digraphs, and list all cyclic strictly Deza digraphs with at most six vertices. Finally, connections to weakly distance-regular digraphs are discussed, and all strictly Deza digraphs which are also weakly distance-regular digraphs with at most ten vertices are listed. [Copyright &y& Elsevier]

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we obtain some examples of weakly distance-regular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distance-regular (respectively weakly distance-transitive) digraphs of girth 2 is characterized. Finally, all commutative weakly distance-regular digraphs of valency 3 and girth 2 are classified. [Copyright &y& Elsevier]

A graph is s-regular if its automorphism group acts regularly on the set of s-arcs. An infinite family of cubic 1-regular graphs was constructed in European J. Combin. 23 (2002) 559, as cyclic coverings of the three-dimensional hypercube Q3. In this paper, we classify the s-regular cyclic coverings of Q3 for each s≥1, whose fibre-preserving automorphism subgroups act arc-transitively. As a result, a new infinite family of cubic 1-regular graphs is constructed. [Copyright &y& Elsevier]

Abstract: It is well known that many famous pooling designs are constructed from mathematical structures by the “containment matrix” method. In this paper, we propose another method and obtain a family of pooling designs with surprisingly high degree of error correction based on a finite set. Given the numbers of items and pools, the error-tolerant property of our designs is much better than that of Maculaʼs designs when the size of the set is large enough. [Copyright &y& Elsevier]

Abstract Let Z p s be the residue class ring of integers modulo p s , where p is a prime number and s is a positive integer. We study subspaces and Grassmann graphs for Z p s n. A Grassmann graph for Z p s n , denoted by G d (n , m , p s) (G d for short), has all m -subspaces of Z p s n as its vertices, and two distinct vertices are adjacent if their intersection is of dimension > m − d , where m < n and 2 ≤ d ≤ m + 1. We give the clique number and geometric structures of maximum cliques of G d , etc. By these results, we obtain the Erdős–Ko–Rado theorem for Z p s n and some bounds of the independence number of G d. We also show that an optimal Grassmannian code for Z p s n is a largest independent set of G d. A p s -Kneser graph over Z p s , denoted by Z p s K n : m where n ≥ 2 m ≥ 4 , is the complement of G m (n , m , p s). Finally, we prove that Z p s K n : m is a core when n > 2 m ≥ 4 , and give some estimations for the chromatic number and clique number of Z p s K n : m. [ABSTRACT FROM AUTHOR]

In this paper, we classify commutative weakly distance-regular digraphs of valency 3 with girth more than 2 and one type of arcs. Combining [8, Theorem 1.2] , [10, Theorem 1.3] and [11, Theorem 1] , commutative weakly distance-regular digraphs of valency 3 are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we obtain upper bounds of the class dimension of flat association schemes in affine and affine-symplectic spaces and construct resolving sets for these schemes. [ABSTRACT FROM AUTHOR]

We describe the full automorphism group of the power (di)graph of a finite group. As an application, we solve a conjecture proposed by Doostabadi, Erfanian and Jafarzadeh in 2013. [ABSTRACT FROM AUTHOR]

The power graph P G of a finite group G is the graph with the vertex set G , where two distinct vertices are adjacent if one is a power of the other. We first show that P G has a transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure of P G . Finally, a closed formula for the metric dimension of P G is established. As an application, we compute the metric dimension of the power graph of a cyclic group. [ABSTRACT FROM AUTHOR]

Abstract: Assume that is a positive integer and or 2. In this paper we introduce the orthogonal graph over a Galois ring of odd characteristic and prove that it is arc transitive. Moreover, we compute its parameters as a quasi-strongly regular graph. In particular, we show that is a strongly regular graph and is a strictly Deza graph when . [Copyright &y& Elsevier]

Let n , r , k 1 , ... , k r and t be positive integers with r ≥ 2 , and F i (1 ≤ i ≤ r) a family of k i -subsets of an n -set V. The families F 1 , F 2 , ... , F r are said to be r -cross t -intersecting if | F 1 ∩ F 2 ∩ ⋯ ∩ F r | ≥ t for all F i ∈ F i (1 ≤ i ≤ r) , and said to be non-trivial if | ∩ 1 ≤ i ≤ r ∩ F ∈ F i F | < t. If the r -cross t -intersecting families F 1 , ... , F r satisfy F 1 = ⋯ = F r = F , then F is well known as r -wise t -intersecting. In this paper, we first describe the structure of maximal 2-cross t -intersecting families with given t -covering numbers and then determine the structure of non-trivial 2-cross t -intersecting families with maximum product of their sizes. We also give a stability result for the non-trivial r -wise t -intersecting families with maximum size for r ≥ 3. [ABSTRACT FROM AUTHOR]

Let P be a symplectic polar space over a finite field F q , and P m denote the set of all m -dimensional subspaces in P. We say a t -intersecting subfamily of P m is trivial if there exists a t -dimensional subspace contained in each member of this family. In this paper, we determine the structure of the maximum sized non-trivial t -intersecting subfamilies of P m. [ABSTRACT FROM AUTHOR]

Abstract: A graph is -regular if its automorphism group acts regularly on the set of its -arcs. In this paper, we classify the -regular elementary Abelian coverings of the three-dimensional hypercube for each whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest one has order 19208. As an application of the classification, all cubic symmetric graphs of order or are classified for each prime , as a continuation of the first two authors’ work, in Y.-Q. Feng, J.H. Kwak [Cubic symmetric graphs of order a small number times a prime or a prime square (submitted for publication)] in which all cubic symmetric graphs of order , , or are classified and of Cheng and Oxley’s classification of symmetric graphs of order , in Y. Cheng, J. Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987) 196–211]. [Copyright &y& Elsevier]

In this paper, we describe the structure of maximal non-trivial uniform t -intersecting families with large size for finite sets. In the special case when t = 1 , our result gives rise to Kostochka and Mubayi's result in 2017. [ABSTRACT FROM AUTHOR]

Let V be an n -dimensional vector space over the finite field F q , and [ V k ] denote the family of all k -dimensional subspaces of V. The families F 1 ⊆ [ V k 1 ] , F 2 ⊆ [ V k 2 ] , ... , F r ⊆ [ V k r ] are said to be r -cross t -intersecting if dim ⁡ (F 1 ∩ F 2 ∩ ⋯ ∩ F r) ≥ t for all F i ∈ F i , 1 ≤ i ≤ r. The r -cross t -intersecting families F 1 , F 2 , ... , F r are said to be non-trivial if dim ⁡ (∩ 1 ≤ i ≤ r ∩ F ∈ F i F) < t. In this paper, we first determine the structure of r -cross t -intersecting families with maximum product of their sizes. As a consequence, we partially prove one of Frankl and Tokushige's conjectures about r -cross 1-intersecting families for vector spaces. Then we describe the structure of non-trivial r -cross t -intersecting families F 1 , F 2 , ... , F r with maximum product of their sizes under the assumptions r = 2 and F 1 = F 2 = ⋯ = F r = F , respectively, where the F in the latter assumption is well known as r -wise t -intersecting family. Meanwhile, stability results for non-trivial r -wise t -intersecting families are also been proved. [ABSTRACT FROM AUTHOR]