Akihiro Munemasa, Kaishun Wang, Yuefeng Yang
We classify certain non-symmetric commutative association schemes. As an application, we determine all the primitive weakly distance-regular circulant digraphs.
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We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself.
Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.
A classification of locally-quasiprimitive circulant digraphs
研究局部拟本原循环有向图的分类，通过扩展局部本原循环图分类，确定其同构类型包括完全图、完全二部图等多种图，给出了连通局部拟本原循环有向图的具体分类结果。
We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one of these families, are normal. That almost all Cayley (di)graphs whose automorphism group is not as small as possible are normal was conjectured by the second author, so these results provide counterexamples to this conjecture. It is then shown that there is a large family of integers for which almost every c…
In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $α(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the lower bounds $α(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and $dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\m…
A note on three-quarters circulant digraphs
本文引入一类与循环群相关的四分之三循环有向图，通过限制生成元组合至三个扇区，利用格点与镶嵌分析可达集增长，给出Moore型上界、显式构造及计算实验，揭示约束格点与极值问题的联系。
A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant digraphs which are integral over a number field K. And we solve the Conjecture3.3 in [XM] and find it is affirmative.
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。
On 2-distance-transitive circulant digraphs
本文基于2-弧传递循环图的分类，完整分类了2-距离传递循环有向图，列出了所有同构类型，包括无向圈、完全多部图、Paley图及若干有向图族。
Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $α$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $α$-d…
$3$-diregular circulant digraph on $12$ vertices is a counterexample of Jackson's conjecture about hamiltonicity of diregular digraphs
A study on Type-2 isomorphic circulant graphs. Part 7: Isomorphism series, digraph and graph of $C_n(R)$
定义循环图的同构集、同构系列、同构有向图与同构图，针对特定阶数循环图给出实例，并发现若干对循环图在模3下既非第一类也非第二类同构。
We solve the long-standing existence problem of vertex-primitive 2-arc-transitive digraphs by constructing an infinite family of such digraphs.
We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.
Path homology of circulant digraphs
本文通过位移自同构和傅里叶分解，提出了一种计算环状有向图路径同调的符号-矩阵方法，揭示了素数与合数n的依赖性及连接集S的稳定性现象。
In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.
In this paper, we study the kth upper bases of primitive non-powerful signed digraphs. A bound on the kth upper bases of all primitive non-powerful signed digraphs is obtained, and the equality case of the bound is characterized. We also show that there exists "gap" in the kth upper base set of primitive non-powerful signed digraphs.
In this paper, we consider the local bases of primitive nonpowerful sign pattern matrices, show that there are "gaps" in the local base set and characterize some sign pattern matrices with given local bases.
The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. The WL-dimension of $Γ$ is defined to be the smallest positive integer $m$ for which $Γ$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We classify the Deza circulant graphs of WL-rank $4$. In additional, it is proved that each of these graphs has WL-dimension at most $3$. Finally, we establish that some families of Deza circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.
Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups
本文针对 Praeger 提出的有限顶点本原 s-弧传递有向图的上界问题，在群为交错群或对称群的情形下证明了 s ≤ 2。
Weakly distance-regular circulants, I
本文分类某些非对称交换结合方案，并借助 Schur 环确定一类弧的全部弱距离正则循环图，同时完成本原情形的分类。
For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least $n$ that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-encounters of length zero), for two particular families of 4-regular directed circulant graphs. We then regard these two families of graphs as families of quantum graphs a…
Periodic orbits on 2-regular circulant digraphs
本文研究了2-正则Cayley有向图上的周期轨道，用步数确定了给定长度的本原周期轨道数量，描述了其格结构，并通过环绕数表征重复数，同时计算了双字母字母表上的Lyndon字数量。
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。
We consider vertex-primitive digraphs having two vertices with almost equal neighbourhoods (that is, the set of vertices that are neighbours of one but not the other is small). We prove a structural result about such digraphs and then apply it to answer a question of Araújo and Cameron about synchronising groups.
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
Vertex-primitive digraphs with large fixity
本文定义了有向图的相对固定性，刻画了相对固定性至少为 $1/3$ 的顶点本原有向图，并证明了有界出度下仅有有限多个此类图。
We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.
Vertex-primitive s-arc-transitive digraphs of symplectic groups
本文针对有限辛群证实了2018年提出的猜想，即有限点本原s-弧传递有向图中的$s$至多为2。
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…
A solution to Babai's problem on digraphs with non-diagonalizable adjacency matrix
本文通过分别构造无限族$s$-弧传递有向图和点本原有向图，解决了Babai关于寻找邻接矩阵不可对角化的有向图的开放问题。
If $G$ is a group acting on a set $Ω$ and $α, β\in Ω$, the digraph whose vertex set is $Ω$ and whose arc set is the orbit $(α, β)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser $G_α$ acting on $Ω$ is called a {\it suborbit} of $G$. A digraph is {\em locally finite} if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph $Γ$ has more than one end if there exists a finite set of vertices $X$ such that the induced digraph $Γ\setminus X$ contains at least two infinite connected components; if there exists such a set containing precis…
On the existence of directed strongly regular graphs with parameters (22, 9, 6, 3, 4)
本文构造了一族参数为 (22, 9, 6, 3, 4) 的有向强正则图，其邻接矩阵由 3×3 循环块构成，自同构群为 Z_3，并用骨架与索具概念刻画其结构。
We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.
We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely the graph $P_1$ with one vertex and no edges. The graph $P_1$ has a unique greatest lower bound, namely the graph $P_2$ with two vertices and one directed edge. Our main result is a complete description of the greatest lower bounds of $P_2$; we call these graphs submaximal. We show that every graph that is not equivalent to $P_1$ and $P_2$ is below one of the submaximal graphs.
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 $s\notin[0,d-2]$…
On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$
本文构建了无限序列的强正则有向图，通过将邻接矩阵表示为由循环块组成的块矩阵，并结合多项式模$x^{2n+3}-1$的紧缩操作，发现稳定的结构模式，从而推导出邻接矩阵的显式公式，并给出了两个此前未解决的图参数例子。
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
Quasi-strongly regular digraphs constructed from transitive groups of degree $n\leq 110$
本文提出一种从传递置换群构造有向正则图的推广方法，证明多组参数的有向强正则图存在，并分类了低次数传递群和本原群产生的拟强正则有向图。
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