arXiv subjects
Publications and source records attributed to Benjian Lv.
Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. 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.
Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.
In this paper, by shifting technique we study $t$-intersecting families for direct products where the ground set is divided into several parts. Assuming the size of each part is sufficiently large, we determine all extremal $t$-intersecting families for direct products. We also prove that every largest $t$-intersecting subfamily of a more general family introduced by Katona is trivial under certain conditions.
In this paper, we study commutative weakly distance-regular digraphs whose attached association schemes are regular, and give a characterization of mixed arcs. As an application, we classify such digraphs of diameter 2.
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Suppose that $\mathscr{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathscr{F}$ is the minimum dimension of a subspace of $V$ which intersects all elements of $\mathscr{F}$. In this paper, we give the tight upper bound for the size of $\mathscr{F}$ whose covering number is $m$, and describe the structure of $\mathscr{F}$ which reaches the upper bound. Moreover, we determine the structure of an maximum intersecting family of singular linear space with the maximum covering number.
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.
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 distanceregular digraphs of valency 3 are completely determined.
A weakly distance-regular digraph is 3-equivalenced if its attached association scheme is 3-equivalenced. In this paper, we classify the family of such digraphs under the assumption of the commutativity.
The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.
Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension formula and dual subspaces on vector subspaces of $\mathbb{Z}^n_{p^s}$. Based on these results, we investigate the Grassmann graph $G_{p^s}(n,m)$ over $\mathbb{Z}_{p^s}$. $G_{p^s}(n,m)$ is a connected vertex-transitive graph, and we determine its valency, clique number and maximum cliques. Finally, we characterize the automorphisms of $G_{p^s}(n,m)$.
A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we show that the valency of any commutative quasi-thin weakly distance-regular digraph is at most 6.
A graph is called a pseudo-core if every endomorphism is either an automorphism or a colouring. In this paper, we show that every Grassmann graph $J_q(n,m)$ is a pseudo-core. Moreover, the Grassmann graph $J_q(n,m)$ is a core whenever $m$ and $n-m+1$ are not relatively prime, and $J_q(2pk-2, pk-1)$ is a core whenever $p,k\geq 2$.
Suzuki (2004) [7] classified thin weakly distance-regular digraphs and pro- posed 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.
In [F. Levstein, C. Maldonado, The Terwilliger algebra of the Johnson schemes, Discrete Math. 307 (2007) 1621--1635], the Terwilliger algebra of the Johnson scheme $J(n,d)$ was determined when $n\geq 3d$. In this paper, we determine the Terwilliger algebra ${\mathcal T}$ for the remaining case $2d\leq n<3d$.
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.
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 $3m\leq n$. The distance-$m$ graph of $J(2m+1,m)$ is the Odd graph $O_{m+1}$. In this paper, we determine the Terwilliger algebra of $O_{m+1}$ and give its basis.
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.
In this note, we prove some combinatorial identities and obtain a simple form of the eigenvalues of $q$-Kneser graphs.