Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

关于偶正则图的R2边连通度的某些结果

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an R2-edge-cut if G-S is disconnected and contains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ″(G)=g(2k-2) for any 2k-regular connected graph G(≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

Transitive tournaments and self‐complementary graphs

A simple proof is given for a result of Sali and Simonyi on self‐complementary graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 111–112, 2001     

Full algebras of matrices

A classification of full algebras of matrices is given. All such algebras are permutation-isomorphic to block lower-triangular matrices with corresponding subdiagonal blocks being either zero-blocks or full. Two full algebras are isomorphic if and only if they are permutation-isomorphic. A one-to-one correspondence is provided between the full algebras and transitive directed graphs. It is also proven that such algebras, if endowed with a lattice order, can be almost f- or d-algebras only if they are diagonal.     

Flag‐transitive 2‐ designs of product type

In this paper, we prove that if a 2‐ design admits a flag‐transitive automorphism group G, then G is of affine, almost simple type, or product type. Furthermore, we prove that if G is product type then is either a 2‐(25, 4, 12) design or a 2‐(25, 4, 18) design with .     

传递图中的限制性边连通度

设G=(V，E)是一个连通图，S包含于E是一个边子集，如果G—S不再连通，且G—S的每一个连通分支都至少含有r个点，则称S为一个r-限制性边割．最小r-限制性边割中所含的边数为G的r-限制性边连通度，记作λ(G)．如果对所有的i=1，…，r，λ(G)都达到其最大可能值，则称G为λ-最优图．王铭和李乔证明了：若G是一个d-正则的点传递图，d≥4，围长g≥5，或者G是一个d-正则的边传递图，d≥4，围长g≥4，则G是λ（g-1)-最优图．本文推广了这一结果，证明了：在同样的条件下，G是λg-最优图．     

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric $F=F(x,y)$ on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature $\mathbf{K}=-1,0,1$ respectively. Especially, for the case when $\mathbf{K}=-1$, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when $\mathbf{K}=1$, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.     

Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in $k^n$, where k is an algebraically closed field of characteristic $p>0$. Let E be a uniform vector bundle over G of rank $r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle $H_d$ or its dual $H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\text{…},d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in $F(d_1,\text{…},d_s)$. Furthermore, we generalize the Grauert–M$\ddot{\mathrm{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.