Junyang Zhang, Sanming Zhou
We correct the statements of two theorems and two corollaries in our paper [On subgroup perfect codes in Cayley graphs, European J. Combin. 91 (2021) 103228]. Proofs of these theorems and three other results are given as well.
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In [Combinatorics, Probability and Computing 16 (2007), 557 - 593, Theorem 1] we proved a polynomial-time bound on the mixing rate of the switch chain for sampling d-regular graphs. This corrigendum corrects a technical error in the proof. In order to fix the error, we must multiply the bound on the mixing time by a factor of d^8 .
The combinatorial heat and wave equations on all finite Cayley and coset graphs with discrete time variable was solved by Lal {\it et al}. In this paper, the results of the above paper are extended for infinite Cayley and coset graphs, whenever the associated groups are discrete, abelian and finitely generated. Furthermore, we study the solution of the combinatorial heat and wave equations on a $k$-regular tree whose associated group is a non-abelian free group on $k$ generators, each of order~$2$. It turns out that in case of Cayley graphs the solutions to combinatorial heat and wave equation…
We give combinatorial generalizations of the Cayley-Bacharach theorem and induced map.
Fourier analysis on distance-regular Cayley graphs over abelian groups
针对任意群类上距离正则Cayley图的刻画问题，本文利用阿贝尔群上的傅里叶分析建立其与有限几何组合对象的联系，结合经典结果分类了群$\mathbb{Z}_n \oplus \mathbb{Z}_p$（$p$为奇素数）上的所有距离正则Cayley图。
A combinatorial proof of the trace Cayley-Hamilton theorem
本文利用方阵对应的加权有向图，通过线性有向子图上的符号加权和解释行列式与特征多项式，并据此给出迹凯莱-哈密顿定理的组合证明，建立特征多项式系数与矩阵幂的迹之间的显式恒等式。
Cayley Incidence Graphs
本文重新表述凯莱图双正则扩展的定义并给出基本性质，揭示其与凯莱超图及陪集几何、差集、笼等几何和组合结构的关联。
Computing the clique number and chromatic number of a general graph are well-known NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. \emph{Linear Algebra Appl.}, 285(1-3): 123--142, 1998) showed that computing clique number and chromatic number are still NP-Hard problems for the class of circulant graphs. We show that computing clique number is NP-Hard for the class of Cayley graphs for the groups $G^n$, where $G$ is any fixed finite group (e.g., cubelike graphs). We a…
We present two hypermatrix formulations of the Cayley Hamilton theorem. One of the proposed formulation naturally extends to hypermatrices the combinatorial interpretations of the classical Cayley Hamilton theorem. We conclude by discussing an application of the theorem to computing graph invariants which distinguish some non-isomorphic graphs with isospectral adjacency matrices.
We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.
Let f be a function mapping an n dimensional vector space over GF(p) to GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence between certain properties of f (e.g., if it is bent) and properties of its associated Cayley graph. Analogously, but much earlier, Dillon showed that f is bent if and only if the "level curves" of f had certain combinatorial properties (again, only when p is 2). The attempt is to investigate an analogous theory when p is greater than 2 using the (apparently new) combinatorial concept of a weighted partial difference set. More precisely, we try…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths. For minimally rigid graphs, the double-exponential Gröbner Bases algorithm with an elimination order can be applied, in theory, but it is impractical even for small instances. By relating the problem to the computation of circuit polynomials in the Cayley-Menger ideal, we recently proposed an algebraic-combinatorial approach and an elimination algorithm for c…
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.
Distance-regular Cayley graphs over (pseudo-) semi-dihedral groups
本文对半二面体群和伪半二面体群上的距离正则 Cayley 图给出部分刻画，这两类群均为含指数 2 循环子群的 2-群，回应了 Miklavič 和 Potočnik 提出的刻画问题。
We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pai…
We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.
Non-Isomorphic Optimal Cayley Graphs
本文构造了七十多个与已知最优Cayley图非同构的度直径最优Cayley图，通过全面比较图不变量，揭示了最优Cayley图的非唯一性，并扩充了已知最优图目录。
Resolving parameters is a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this article, we construct a class of Toeplitz graphs, and will be denoted by $T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs are vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, a…
Binomial Cayley Graphs and Applications to Dynamics on Finite Spaces
本文引入二项式凯莱图，通过邻接矩阵谱分析研究其组合性质，并借助 RSK 对应将对称群情形下的零特征值重数与最长递增子序列联系起来，进而用于描述粒子-盒子系统的简并性。
Eternal domination in Cayley graphs
研究凯莱图中永恒支配问题，通过证明顶点传递图、广义二面体群上的凯莱图等相关性质，给出永恒支配数与支配数关系的结论及不同情况的无限族图。
We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\le r \le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).
Distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$
本文解决了Miklavič和Potočnik提出的距离正则Cayley图刻画问题，完全确定了奇素数p下 $\bb Z_{p^s}\bb Z_{p}$ 上所有距离正则Cayley图的同构类型，为相关代数图论研究提供了分类结果。
Splines on Cayley Graphs of the Symmetric Group
本文研究对称群Cayley图上的样条环，推广GKM构造，给出极小生成集下的显式生成元、分次平凡性的组合刻画，并计算首项分次特征。
On uniquely colorable Cayley graphs
该研究解决了Klotz和Sander提出的两个关于唯一可着色Cayley图的开放问题，构造了无限族唯一3-可着色整循环图并否定相关猜想，还建立通用代数构造证明非阿贝尔群上存在符合要求的唯一可着色Cayley图。
A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism g…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there ar…
In this paper we study the length of the longest induced cycle in the unitary Cayley graph $X_n = Cay(\mathbb Z_n; U_n)$, where $U_n$ is the group of units in $\mathbb Z_n$. Using residues modulo the primes dividing $n$, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing $n$, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in $X_n$. We also see that if $n$ has…
Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group S_n generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The purpose of this note is to point out that, as mentioned also by P. Renteln, this problem is actually already solved in another context. Indeed it is equivalent to studying the spectrum of so-called Jucys-Murphy elements in the algebra of the symmetric group, which is well understood. The aforementioned conjectures are direct consequences of the existing theor…
Let $Γ$ be a finite graph and let $A(Γ)$ be its adjacency matrix. Then $Γ$ is {\it singular} if $A(Γ)$ is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs ${\rm Cay}(G,H)$ when $G$ is a finite group and when the connecting set $H$ is a union of conjugacy classes of $G.$ In this situation the singularity problem reduces to finding an irreducible character $χ$ of $G$ for which $\sum_{h\in H}\,χ(h)=0.$ At this stage we focus on the case when $H$ is a single conjugacy class $h^G$ of $G.$ Here …
Factor maps for automorphism groups via Cayley diagrams
本文利用群作用与边标记图的对应关系统一了弱包含、局部-全局收敛与连续模型论的相关结果，并系统研究了 Aut(Cay(Γ))-fiid 与 Γ-fiid 组合学的差异，通过构造例子与反例刻画了哪些群的凯莱图允许（近似）fiid 凯莱图。
Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius Groups
本文对比例一弗罗贝尼乌斯群$G=N\rtimes H$上的正规连接集拉马努金凯莱图进行分类，利用膨胀现象将其转化为补群$H$上的图分类，最终得到适用于$\text{AGL}(1,q)$等的完整分类结果。
We show that any connected Cayley graph $Γ$ on an Abelian group of order $2n$ and degree $\tildeΩ(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Γ$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.
This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We …
Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.
Explicit Fractalizers: Prime-Order Paley Graphs and other Cayley Graphs
本文证明每个足够大的素数阶 Paley 图是分形图，并构造了第二个显式 Cayley 分形图无限族，解决了极值组合学中显式构造问题。
Abelian groups without 3-chromatic Cayley graphs
本文研究了交换群中是否存在3染色的Cayley图，证明了当群的指数不为1、2或4时存在3染色的Cayley图，并进一步得出关于图色数的拓扑结果。
Stability of Cayley graphs and Schur rings
本文通过Schur环理论刻画了连通非二部凯莱图不稳定的充要条件，并应用于奇数阶阿贝尔群凯莱图的孪生顶点性质及阶为2p^e的循环图不稳定性判定。
The splitting field $\mathbb{SF}(Γ)$ of a mixed graph $Γ$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $Γ$. The extension degree $[\mathbb{SF}(Γ):\mathbb{Q}]$ is called the algebraic degree of $Γ$. In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of [K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups…
Asymptotic stability of Cayley graphs on abelian groups
本文证明几乎所有有限阿贝尔群的赛雷图都是稳定的，通过研究其常规双覆盖的自同构群最小化问题。
The $(n,k)$-star graphs are an important class of interconnection networks that generalize star graphs, which are superior to hypercubes. In this paper, we continue the work begun by Cheng et al.~(Graphs and Combinatorics 2017) and complete the classification of all the $(n,k)$-star graphs that are Cayley.
Perfect state transfer in Grover walks on normal Cayley graphs
研究正规凯莱图上格罗弗游走的完美状态转移，建立充要条件，推导阿贝尔群等上凯莱图的谱准则及组合特征，给出酉凯莱图完美状态转移完整特征，证明其中恰有四个图展示该转移。
What is and is not inside a Cayley graph?
本文证明存在非任何最小Cayley图子图的围长5三次图，但满足gcd(n,k)=1的广义Petersen图可为最小Cayley图的诱导子图，回应Babai的评论。
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a full connected component of a Cayley tree and for the second case the polygon contain two fixed vertices. From these formulas, which is in the form of finite linear combination of Catalan numbers, one can find the asymptotic estimation for a counting problem.
Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept of directed strongly regular graphs. In this paper we construct several rich families of directed strongly regular graphs with new parameters. Our constructions yielding new parameters are based on extending known explicit constructions to cover more parameter sets. We also explore some of the links between Cayley graphs, block matrices and directed strongly regular graphs with certain parameters. Directed strongly regular graphs which are…
On Some Bi-Cayley Graphs over Cyclic Groups of Order $p^2 q^2$ and Related Extensions
本文研究了阶为$p^2 q^2$的循环群上的双Cayley图的结构和性质，证明其连通性、双正则性及环长为三，并扩展了相关结果到更一般的有限群情况。
Cayley trees and increasing 1,2-trees: let's twist!
本文证明递增1,2-树与Cayley树计数相同，其中三角形数对应扭转数，并给出三个证明及高效随机生成器。
We discuss several combinatorial problems that arise when one looks at computational algorithms for highly symmetric networks of processors. More specifically, we are interested in minimal times associated with four communication tasks (defined more precisely below): universal broadcast, every processor has a vector that it wishes to broadcast to all the others; universal accumulation, every processor wishes to receive the sum of all the vectors being sent to it by all the other processors; universal exchange, every processor wishes to exchange a vector with each other processor; and global su…
Cayley graphs of quasirandom groups
研究拟随机群中扩张性质与简单商群的关系，通过移除Bourgain-Gamburd扩张机假设中的乘积定理，并利用冠理论证明少量随机元素高概率生成扩张Cayley图，以及拟随机群Cayley图直径的多对数上界。
Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$
本研究聚焦ℱ₂ʳ的Cayley图沙堆群中研究不足的Sylow-2分量，借助表示理论的环结构与进位组合学计算二项式和2进赋值，得出一般图Sylow-2循环因子数量、非一般图对应界值及最大循环因子精确上界，还给出超立方体情形下最大n-1个Sylow-2循环因子的精确公式。
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\geq4$. Even for this problem, it was only solved f…
A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs
该研究对有限阿贝尔群上表现出完美状态转移的连通凯莱图进行完全分类，确定其度数为1、2或4，给出度数为4时的充要条件，并将相关循环图分类结果扩展至任意有限阿贝尔群上的所有度数凯莱图。
The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups
本文证明对称群上由移动前r个点的k-循环生成的非正规Cayley图具有Aldous性质，即严格第二大特征值由标准表示取得，仅排除两类例外情况，并确定了相关图的特征值对应的不可约表示。
On the Iwasawa theory of Cayley graphs
本文借助表示论研究 Cayley 图在 $\mathbb{Z}_\ell$-塔中的 Iwasawa 不变量，给出 Iwasawa 多项式因式分解与 $μ$、$λ$ 不变量分解，并在完全图上建立其消失条件。
Odd cycles in symmetric Cayley graphs on prime cyclic groups
本文研究了素数阶循环群上对称Cayley图中避免固定长度奇圈的最大生成集大小，通过加性零和奇围长论证与弱奇泛圈性，给出了精确极值并构造了极值族。
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number $d$ for which one can label each vertex with a nonzero $d$-dimensional co…
On the gradient of the coefficient of the characteristic polynomial
本文证明双变量凯莱-哈密顿定理，统一导出经典凯莱-哈密顿定理、吉拉德-牛顿恒等式及行列式与特征多项式系数的多项式规模ABP，并基于系数梯度构造出规模为原纪录三分之一、宽度一半的新ABP，首次在28年内改进该问题，同时回答了clow序列必要性的26年未决问题。
CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs
本文提出一种结合强化学习与扩散距离方法的新型路径寻找方法，通过基准测试和数学方法验证了对称群Cayley图直径的猜想，并在Kaggle平台发起挑战以促进众包活动。
The List-distinguishing chromatic number of graphs containing only small complete bigraphs
本文受Catlin组合技术启发，为仅含小完全二部图导出子图的连通图类建立了可区别色数与列表可区别色数的改进上界，并应用Menger定理展示了结果在基于Paley图、Dihedral群Cayley图等构造的图中的应用。
On the clique number of random Cayley graphs and related topics
本文用纯组合方法证明随机凯莱图的团数高概率为 O(log N log log N)，改进了 Alon 的上界，并据此证明 Alon 关于凯莱图团数与独立数均为 O(log N) 的猜想对几乎所有阿贝尔群成立，还构造了自补凯莱图以强形式解决 Alon–Orlitsky 问题。
Symmetric Cayley graphs on non-abelian simple groups of valency 7
该研究改进了非正规7度对称Cayley图的已有刻画，证明其自同构群含特定弧传递非交换单正规子群，明确了群对结构与最大可解正规子群相关性质。
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