arXiv subjects
Publications and source records attributed to Kaishun Wang.
Levstein and Maldonado [F. Levstein, C. Maldonado, The Terwilliger algebra of the Johnson schemes, Discrete Mathematics 307 (2007) 1621--1635] computed the Terwilliger algebra of the Johnson scheme $J(n,m)$ when $3m\leq n$. In this paper, we determine the Terwilliger algebra of the incidence graph $J(n,m,m+1)$ of Johnson geometry when $3m\leq n$, give two bases of this algebra, and calculate its dimension.
Let $G$ be a (di)graph. A set $W$ of vertices in $G$ is a \emph{resolving set} of $G$ if every vertex $u$ of $G$ is uniquely determined by its vector of distances to all the vertices in $W$. The \emph{metric dimension} $μ(G)$ of $G$ is the minimum cardinality of all the resolving sets of $G$. Cáceres et al. \cite{Ca2} computed the metric dimension of the line graphs of complete bipartite graphs. Recently, Bailey and Cameron \cite{Ba} computed the metric dimension of the line graphs of complete graphs. In this paper we study the metric dimension of the line graph $L(G)$ of $G$. In particular, we show that $μ(L(G))=|E(G)|-|V(G)|$ for a strongly connected digraph $G$ except for directed cycles, where $V(G)$ is the vertex set and $E(G)$ is the edge set of $G$. As a corollary, the metric dimension of de Brujin digraphs and Kautz digraphs is given. Moreover, we prove that $\lceil\log_2Δ(G)\rceil\leqμ(L(G))\leq |V(G)|-2$ for a simple connected graph $G$ with at least five vertices, where $Δ(G)$ is the maximum degree of $G$. Finally, we obtain the metric dimension of the line graph of a tree in terms of its parameters.
Let $S_n$ be the symmetric group on $n$ points. Deza and Frankl [M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352--360] proved that if ${\cal F}$ is an intersecting set in $S_n$ then $|{\cal F}|\leq(n-1)!$. In this paper we consider the $q$-analogue version of this result. Let $\mathbb{F}_q^n$ be the $n$-dimensional row vector space over a finite field $\mathbb{F}_q$ and $GL_n(\mathbb{F}_q)$ the general linear group of degree $n$. A set ${\cal F}_q\subseteq GL_n(\mathbb{F}_q)$ is {\it intersecting} if for any $T,S\in{\cal F}_q$ there exists a non-zero vector $α\in \mathbb{F}_q^n$ such that $αT=αS$. Let ${\cal F}_q$ be an intersecting set in $GL_n(\mathbb{F}_q)$. We show that $|{\cal F}_q|\leq q^{(n-1)n/2}\prod_{i=1}^{n-1}(q^i-1)$.
n [D. de Caen, E.R. van Dam. Fissioned triangular schemes via the cross-ratio, {Europ. J. Combin.}, 22 (2001) 297-301], de Caen and van Dam constructed a fission scheme $\FT(q+1)$ of the triangular scheme on $\PG(1,q)$. This fission scheme comes from the naturally induced action of $\PGL(2,q)$ on the 2-element subsets of $\PG(1,q)$. The group $\PGL(2,q)$ is one of two infinite families of finite sharply 3-transitive groups. The other such family $\Mq(q)$ is a "twisted" version of $\PGL(2,q)$, where $q$ is an even power of an odd prime. The group $\PSL(2,q)$ is the intersection of $\PGL(2,q)$ and $\Mq(q)$. In this paper, we investigate the association schemes coming from the actions of $\PSL(2,q)$, $\Mq(q)$ and $\PML(2,q)$, respectively. Through the conic model introduced in [H.D.L. Hollmann, Q. Xiang. Association schemes from the actions of $\PGL(2, q) $ fixing a nonsingular conic, {J. Algebraic Combin.}, 24 (2006) 157-193], we introduce an embedding of $\PML(2,q)$ into $\PML(3,q)$. For each of the three groups mentioned above, this embedding produces two more isomorphic association schemes: one on hyperbolic lines and the other on hyperbolic points (via a null parity) in a 3-dimensional orthogonal geometry. This embedding enables us to treat these three isomorphic association schemes simultaneously.
For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. \cite{Jiang} showed that the minimum number of vertices of realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Kr$\acute{\rm a}$l \cite{Kral} proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we improve Kr$\acute{\rm a}$l's result, and determine the size of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Kr$\acute{\rm a}$l in 2004.
Gravier et al. investigated the identifying codes of Cartesian product of two graphs. In this paper we consider the identifying codes of lexicographic product G[H] of a connected graph G and an arbitrary graph H, and obtain the minimum cardinality of identifying codes of G[H] in terms of some parameters of G and H.
As a generalization of singular linear spaces, we introduce the concept of t-singular linear spaces, make some anzahl formulas of subspaces, and determine the suborbits of t-singular linear groups.
As one of the serial papers on suborbits of point stabilizers in classical groups on the last subconstituent of dual polar graphs, the corresponding problem for orthogonal dual polar graphs over a finite field of odd characteristic is discussed in this paper. We determine all the suborbits of a point-stabilizer in the orthogonal group on the last subconstituent, and calculate the length of each suborbit. Moreover, we discuss the quasi-strongly regular graphs and the association schemes based on the last subconstituent, respectively.
The metric dimension of a graph is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. Bailey and Meagher obtained an upper bound on the metric dimension of Grassmann graphs. In this paper we obtain an upper bound on the metric dimension of bilinear forms graphs.
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 Macula's designs when the size of the set is large enough.
Pooling designs are standard experimental tools in many biotechnical applications. It is well-known that all famous pooling designs are constructed from mathematical structures by the "containment matrix" method. In particular, Macula's designs (resp. Ngo and Du's designs) are constructed by the containment relation of subsets (resp. subspaces) in a finite set (resp. vector space). Recently, we generalized Macula's designs and obtained a family of pooling designs with more high degree of error correction by subsets in a finite set. In this paper, as a generalization of Ngo and Du's designs, we study the corresponding problems in a finite vector space and obtain a family of pooling designs with surprisingly high degree of error correction. Our designs and Ngo and Du's designs have the same number of items and pools, respectively, but the error-tolerant property is much better than that of Ngo and Du's designs, which was given by D'yachkov et al. \cite{DF}, when the dimension of the space is large enough.
Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and $r_{n_i}=s_i, i=1,2,...,t$, where each $r_{n_i}$ is the $n_i$th component of the chromatic spectrum of ${\cal H}$. As a result, we solve one open problem for 3-uniform bi-hypergraphs proposed by Bujtás and Tuza in 2008. Moreover, we find a family of sub-hypergraphs with the same feasible set and the same chromatic spectrum as it's own. In particular, we obtain a small upper bound on the minimum number of vertices in 3-uniform bi-hypergraphs with any given feasible set.
In this note, we prove some combinatorial identities and obtain a simple form of the eigenvalues of $q$-Kneser graphs.
A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd graphs, doubled Grassmann graphs and twisted Grassmann graphs, respectively, and obtain the upper bounds on the metric dimension of these graphs.
An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers 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 [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.