Min Sha
We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.
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Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
本文研究 $0$-单位 $3$-度循环有向图的自同态幺半群的自同构群，通过建立无 $2$-圈单位循环有向图存在诱导有向圈的自同态的充要条件，显式确定了正规化子群 $U_S(\mathbb{Z}_n)$，解决了该族的未决问题。
The Kautz digraphs $K(d,\ell)$ are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related to these, the cyclic Kautz digraphs $CK(d,\ell)$ were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs $sK(d,\ell)$, from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices…
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs $MCK(d,\ell)$ and it is derived from the Kautz digraphs $K(d,\ell)$. %It is not common to find non-regular digraphs with minimal diameter given their number of vertices and out-degree. It is well-known that the Kautz digraphs $K(d,\ell)$ h…
Cyclic Neighborhoods in Digraphs
针对满足任意顶点邻域非循环的有向图，证明强连通时边数下界为7n/3、强2-连通时为8n/3，两个下界均为最优。
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $σ\in Aut(G)$ such that $S^σ=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if…
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
Finite smooth digraphs, that is, finite directed graphs without sources and sinks, can be partially ordered via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two smooth digraphs in our poset it suffices to show that the polymorphism clone of one of the digraphs satisfies a prime cyclic loop condition that is not satisfied by the polymorphism clone of the other. Furthermore, we prove that the poset of cyclic loop ordered by their strength for clones is a distributive l…
Let $Γ=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(Γ)$. The Cayley index of $Γ$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $Γ$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $Γ$ on $G$ whose …
We define the family of {\it locally path-bounded} digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible, in a finite number of moves. This is done by proving that the Generalized Sprague-Grundy function exists uniquely and has finite values on this class.
Normal and non-normal Cayley digraphs on cyclic and dihedral groups
本文研究循环群与二面体群上兼具正规性及非正规正则同构子群的 Cayley 有向图，证明循环群均不构成 NNND/NNN-群，并给出二面体群的完整阶数判定。
We say that a digraph is essentially cyclic if its Laplacian spectrum is not completely real. The essential cyclicity implies the presence of directed cycles, but not vice versa. The problem of characterizing essential cyclicity in terms of graph topology is difficult and yet unsolved. Its solution is important for some applications of graph theory, including that in decentralized control. In the present paper, this problem is solved with respect to the class of digraphs with ring structure, which models some typical communication networks. It is shown that the digraphs in this class are essen…
Digraph-defined external difference families and new circular external difference families
提出基于图和有向图的统一框架研究外部差族（EDFs），利用有限域割圆术和循环群直接构造获得新型循环外部差族（CEDFs）的显式构造，并首次在非循环阿贝尔群和非阿贝尔群中构造出CEDF。
Cyclic structure, vertex degree and number of linear vertices in Minimal Strong Digraphs
本研究针对极小强有向图（MSDs），推导了线性顶点数与顶点度的关系、最长环长度的上界，拓展了其特征多项式系数的界，并证明MSD最长环计算问题为NP-Hard。
We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).
Endomorphism and automorphism graphs of finite groups
本文定义有限群的有向/无向自同态图与自同构图，介绍其与变换幺半群的关联，探究其图论性质、判定对应图类的群类型并计算特殊群实例，最后以示例说明相关问题的复杂性。
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedr…
A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure. In this paper we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectional…
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $Γ(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$…
Towards inductive proofs in algebraic combinatorics
本文引入了一类新的传递置换群，给出了其商保持在该族中的充分条件，并利用归纳法确定了阶为$p^n$的循环有向图的自同构群。
On the Adjacency spectra of alternating-oriented $n$-gonal staircase digraphs
研究了交替定向n边阶梯有向图的邻接谱，通过分析其邻接矩阵的结构和性质，确定了非零特征值的分布和重数，并与Padovan螺旋数相关联。
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $α\in Aut(G)$ such that $S^α=T$, whenever $Cay(G,S)\cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture …
New classes of groups related to algebraic combinatorics with applications to isomorphism problems
本文引入$5/2$-群类的两个精化，证明其正则循环子群共轭，从而推广并简化了对称构型与单位循环有向图同构的判定证明。
Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs
证明了有限阿贝尔群上两个不同非零生成元的Cayley有向图存在两条弧不交哈密顿路径，通过切割反射定理和扇区填充不等式解决。
Zeta Functions for Spherical Tits Buildings of Finite General Linear Groups
本文为有限一般线性群相关的球面建筑定义了边zeta函数，通过引入有向图X0和X2、循环n部图、部传递群作用及Hecke代数的Springer定理，推导出优雅公式并揭示特征值模式。
We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\sqrt{n}$, the `large' exponents $\lfloor \frac{n}{3} \rfloor +1, \lfloor \frac{n}{2} \rfloor, n-1$, and for even $n \ge 4$, the additional value $\frac{n}{2}-1$. It is easy to see that no exponent in $[\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\lfloor \frac{n}{3}\rfloor +2,\frac{n}{2}-2]$ is possible. Extend…
Cyclic Shuffle Groups: Universal Two-Transitivity and Complete Classification
该论文研究由k堆完美洗牌与循环堆置换生成的群H_{k,n}，证明其在n非k幂时为2-传递，完成H_{k,n}及含C_k的堆群P对应的Sh(P,n)的完全分类，解决相关群论猜想。
Optimal bounds for zero-sum cycles. I. Odd order
该研究属于组合数学领域，针对有限群上完全有向图的零和环最小阶参数$n(Γ)$，证明了奇数阶群满足最优界$n(Γ)\le |Γ|+1$，确定循环群的$n(\mathbb{Z}_q)=q+1$，偶数阶情形将在续篇给出。
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