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Weijun Liu

Publications and source records attributed to Weijun Liu.

18 recordsLinked to original sources
arXiv2026

Large maximal subgroups of almost simple classical groups

A proper subgroup \(H\) of a finite group \(G\) is called large if \(|H|^3 \ge |G|\). This paper characterises all the large maximal subgroups of almost simple groups whose socle is a finite simple classical group. For such a socle \(G_0\) and a core-free subgroup \(H_0\), define \(\mathcal{Q}(G_0,H_0)\) as the set of almost simple groups \(G\) with socle \(G_0\) that have a large maximal subgroup \(H\) satisfying \(H_0 = H \cap G_0\). We establish necessary and sufficient conditions on the pair $(G_0,H_0)$ for the set \(\mathcal{Q}(G_0,H_0)\) to be nonempty.

math.GR↗
arXiv2024

On biprimitive semisymmetric graphs

A regular bipartite graph $Γ$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(Γ)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(Γ)$ that stabilizes the biparts of $Γ$, we say that $Γ$ is $G$-biprimitive if $G$ acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of $G$-biprimitive semisymmetric graphs is obtained for $G=\mathrm{A}_n$ or $\mathrm{S}_n$. In pursuit of this goal, we determine all pairs of maximal subgroups of $\mathrm{A}_n$ or $\mathrm{S}_n$ with the same order and all pairs of almost simple groups of the same order.

math.GR↗
arXiv2023

Block-transitive $3$-$(v,k,1)$ designs on exceptional groups of Lie type

Let $\mathcal{D}$ be a non-trivial $G$-block-transitive $3$-$(v,k,1)$ design, where $T\leq G \leq \mathrm{Aut}(T)$ for some finite non-abelian simple group $T$. It is proved that if $T$ is a simple exceptional group of Lie type, then $T$ is either the Suzuki group ${}^2B_2(q)$ or $G_2(q)$. Furthermore, if $T={}^2B_2(q)$ then the design $\mathcal{D}$ has parameters $v=q^2+1$ and $k=q+1$, and so $\mathcal{D}$ is an inverse plane of order $q$; and if $T=G_2(q)$ then the point stabilizer in $T$ is either $\mathrm{SL}_3(q).2$ or $\mathrm{SU}_3(q).2$, and the parameter $k$ satisfies very restricted conditions.

math.CO↗
arXiv2023

Block-transitive $3$-$(v,k,1)$ designs associated with alternating groups

Let $\mathcal{D}$ be a nontrivial $3$-$(v,k,1)$ design admitting a block-transitive group $G$ of automorphisms. A recent work of Gan and the second author asserts that $G$ is either affine or almost simple. In this paper, it is proved that if $G$ is almost simple with socle an alternating group, then $\mathcal{D}$ is the unique $3$-$(10,4,1)$ design, and $G=\mathrm{PGL}(2,9)$, $\mathrm{M}_{10}$ or $\mathrm{Aut}(\mathrm{A}_6 )=\mathrm{S}_6:\mathrm{Z}_2$, and $G$ is flag-transitive.

math.GR↗
arXiv2022

Inequalities for generalized matrix function and inner product

We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces, and then provide a unified extension of the recent inequalities due to Lin [Electron. J. Linear Algebra 27 (2014) 821-826], Zhang et al. [Linear Algebra Appl. 498 (2016) 99-105] and [Electron. J. Linear Algebra 27 (2014) 332-341] and a result of Choi [Linear Algebra Appl. 532 (2017) 1-7]. Moreover, we demonstrate the application of a positive semidefinite $3\times 3$ block matrix, which motivates us to give alternative proofs of Dragomir's inequality and Krein's inequality.

math.FA↗
arXiv2022

A survey on spectral conditions for some extremal graph problems

This survey is two-fold. We first report new progress on the spectral extremal results on the Turán type problems in graph theory. More precisely, we shall summarize the spectral Turán function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By a unified method, we present some sufficient conditions based on the adjacency spectral radius and the signless Laplacian spectral radius for a graph to be Hamiltonian, $k$-Hamiltonian, $k$-edge-Hamiltonian, traceable, $k$-path-coverable, $k$-connected, $k$-edge-connected, Hamilton-connected, perfect matching and $β$-deficient.

math.CO↗
arXiv2022

Spectral radius and the $2$-power of Hamilton cycles

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $μ(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without containing the $2$-power of a Hamilton cycle.

math.CO↗
arXiv2021

A new matrix inequality involving partial traces

Let $A$ be an $m\times m$ positive semidefinite block matrix with each block being $n$-square. We write $\mathrm{tr}_1$ and $\mathrm{tr}_2$ for the first and second partial trace, respectively. In this paper, we prove the following inequality \[ (\mathrm{tr} A)I_{mn} - (\mathrm{tr}_2 A) \otimes I_n \ge \pm \bigl( I_m\otimes (\mathrm{tr}_1 A) -A\bigr).\] This inequality is not only a generalization of Ando's result [ILAS Conference (2014)] and Lin [Canad. Math. Bull. 59 (2016) 585--591], but it also could be regarded as a complement of a recent result of Choi [Linear Multilinear Algebra 66 (2018) 1619--1625]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.

math.FA↗
arXiv2021

Block-transitive, point-primitive Steiner 3-designs

This paper studies the long-standing open problem of the reduction of Steiner 3-designs admitting a block-transitive automorphism group. We prove that if G acts as a point-primitive, block-transitive automorphism group of a nontrivial Steiner 3-design, then G is of affine or almost simple type.

math.CO↗
arXiv2021

The Q-index and connectivity of graphs

A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficient condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.

math.CO↗
arXiv2020

Another determinantal inequality involving partial traces

Let $A$ be a positive semidefinite $m\times m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds \[ (\mathrm{tr} A)^{mn} - \det(\mathrm{tr}_2 A)^n \ge \bigl| \det A - \det(\mathrm{tr}_1 A)^m \bigr|, \] where $\mathrm{tr}_1$ and $\mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].

math.FA↗
arXiv2020

Some applications of two completely copositive maps

A linear map $Φ:\mathbb{M}_n \to \mathbb{M}_k$ is called completely copositive if the resulting matrix $[Φ(A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $Φ(X)=(\mathrm{tr} X)I+X$ and $Ψ(X)= (\mathrm{tr} X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3\times 3$ block matrices are included.

math.FA↗
arXiv2019

Some properties for morphism of representations

Let $ψ: G\to GL(V)$ and $φ:G \to GL (W)$ be representations of finite group $G$. A linear map $T: V\to W$ is called a morphism from $ψ$ to $φ$ if it satisfys $Tψ_g= φ_g T$ for each $g\in G$ and let $\mathrm{Hom}_G (ψ,φ)$ denote the set of all morphisms. In this paper, we make full stufy of the subspace $\mathrm{Hom}_G(ψ, φ)$. As byproducts, we include the proof of the first orthogonality relation and Schur's orthogonality relation.

math.RT↗