Alexander Lubotzky, Beth Samuels, Uzi Vishne
We apply spectral analysis of quotients of the Bruhat-Tits buildings of type $\tilde{A}_{d-1}$ to construct isospectral non-isomorphic Cayley graphs of the finite simple groups $\operatorname{PSL}_d({\mathbb F}_q)$ for every $d \geq 5$ ($d \neq 6$) and prime power $q > 2$.
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We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ and any symmetric subset $S$. We then focus on the family of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples $\{G_R, G_R^+, \b…
Isospectral Cayley graphs with even and odd spectrum
本文研究了镜像双Cayley图的谱性质，发现当T属于特定集合时，图的谱为偶或奇，从而构造了偶谱和奇谱的等谱Cayley图对。
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley graphs of cyclic groups with the same list of eigenvalues of their adjacency matrices will be presented.
The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and Vishne proved that there are isospectral non-isomorphic Cayley graphs of $\mathrm{PSL}_d(\mathbb F_q)$ for every $d\geq 5$ ($d \neq 6$) and prime power $q> 2$. In this article, we focus on three variants of Cayley graphs, viz., the Cayley sum graphs, the twisted Cayley graphs, and the twisted Cayley sum graphs. We prove the existence of non-isomorphic expande…
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.
The energy of a family of mirror di-Cayley (sum) graphs: equienergy and moments
本文计算镜像双Cayley（和）图的能量与谱k-矩，研究其亚能量、序能量和超能量性质，并给出非同构等能量对存在的条件。
We give a new formula for computing the isospectral reduction of a matrix (and graph) down to a submatrix (or subgraph). Using this, we generalize the notion of isospectral reductions. In addition, we give a procedure for constructing a matrix whose isospectral reduction down to a submatrix is given. We also prove that the isospectral reduction completely determines the restriction of the quantum walk transition matrix to a subset. Using these, we construct new families of simple graphs exhibiting perfect quantum state transfer.
Simple Cayley permutations
本文提出与膨胀相容的Cayley排列简单性概念，证明其替换分解，并据此枚举三类简单对象，同时证明有限简单成员的遗传类具有有限基和代数生成函数。
We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta function to digraphs.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.
In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let $\Ga$ be such a Cayley graph on a non-abelian simple group $T$. It is proved that either $\Ga$ is a normal Cayley graph or $\Ga$ is $S$-arc-transitive, with $(S,T)=(\A_n,\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on $\A_{n-1}$ and specific examples are constructed.
A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if $G$ is normal in $\Aut\Ga$. The concept of normal Cayley graphs was first proposed by M.Y.Xu in [Discrete Math. 182, 309-319, 1998] and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we investigate the normality problem of the connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups. We prove that all such graphs $\Ga$ are either normal or $G=\A_{39}$ or $\A_{79}$. Further, a connected arc-transitive pentavalent Cayley graph…
We present a nice result on the probability of a cycle occurring in a randomly generated graph. We then provide some extensions and applications, including the proof of the famous Cayley formula, which states that the number of labeled trees on $n$ vertices is $n^{n-2}.$
Prime-valent Symmetric Cayley Graphs of Characteristically Simple Groups
本文刻画了特征单群$T^k$（$k\geq2$）的连通本原度弧传递Cayley图的结构，并证明三次情形下所有连通对称Cayley图均为正规图。
Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symme…
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.
On Cayley representations of central Cayley graphs over almost simple groups
本文研究单群与拟单群上中心Cayley图的Cayley表示，证明单群上至多有两个不等价表示，并给出拟单群情形下多项式时间求全部不等价表示的算法。
We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertex-transitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle language, or just the same elementary cycle language. Under the assumption of the axiom of choice, we characterize the Cayley graphs for all group subsets as the deterministic, co-deterministic, vertex-transitive simple graphs.
An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study of $2$-arc-transitive Cayley graphs on $\mathrm{A}_n$ for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete $(G,2)$-arc-transitive Cayley graph on $\mathrm{A}_n$ with $G$ almost simple, the socle of $G$ is either $\mathrm{A}_{n+1}$ or $\mathrm{A}_{n+2}$. We also construct …
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.
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 …
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that its automorphism group ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the partite sets of $Σ$. It is well-known that every Haar graph of finite abelian groups is a Cayley graph. In this paper, we prove that every finite non-abelian group admits a non-Cayley Haar graph except the dihedral groups $D_6$, $D_…
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
A graph $Γ$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(Γ)$ is transitive on the set of vertices of $Γ$ and the set of $s$-arcs of $Γ$, where for an integer $s \ge 1$ an $s$-arc of $Γ$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $Γ$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $Γ$ is called 2-transitive if it is $(\mathrm{Aut}(Γ), 2)$-arc-transitive but not $(\mathrm{Aut}(Γ), 3)$-arc-transitive. A Cayley graph $Γ$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(Γ)$ and non-normal othe…
We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.
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 Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner…
In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over $D_n$ in terms of the Boolean algebra of $\langle a\rangle$, from which we find infinite classes of integral Cayley graphs over $D_n$. In particular, we completely determine all integral Cayley graphs over the dihedral group $D_p$ for a prime $p$.
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.
Li and Praeger classified finite nonabelian simple groups, it has only one or two fusion classes of any certain value. As a by-product, they classified m-CI-groups, which is critical in the research of Cayley graphs. In the paper, we will consider generalized Cayley graphs. This concept is proposed by Marusic et al. In the paper, (local) m- GCI-group is defined, and we get many properties and characterizations based on the generalized Cayley isomorphism, which are the key measures for the classification of (local) m-GCI-group. And above all, we will give a classification of local 2-GCI-groups …
Nowhere-zero $3$-flows in Cayley graphs on solvable groups of twice square-free order
本文验证了Tutte的3-流猜想，针对Cayley图在可解群二次平方自由阶上的情况，提出新的必要充分条件以确定简单5度图是否具有无零3-流。
In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.
Given a finite simple graph $\cG$ with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of $\cG$, interpreted as transpositions. We show that, if $\cG$ is complete multipartite, the eigenvalues of the Laplacian of $\Cay(\cG)$ have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of $\cG$ and of $\Cay(\cG)$ have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that th…
Perfect state transfer (PST) has great significance due to its applications in quantum information processing and quantum computation. In this paper we present a characterization on connected simple Cayley graph $Γ={\rm Cay}(G,S)$ having PST. We show that many previous results on periodicity and existence of PST of circulant graphs (where the underlying group $G$ is cyclic) and cubelike graphs ($G=(\mathbb{F}_2^n,+)$) can be derived or generalized to arbitrary abelian case in unified and more simple ways from our characterization. We also get several new results including answers on some probl…
We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
On symmetric Cayley graphs of valency thirteen
本文研究了有限非交换单群上连通13度对称Cayley图的正规性问题，证明了其除少数特定交错群外均正规，并构造了新的非正规图例。
In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of…
A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. \cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.
A complete classification of perfect unitary Cayley graphs
本文研究酉凯莱图的完美性，并给出了与有限环相关的完美酉凯莱图的完全分类。
Total Coloring for some classes of Cayley graphs
本文研究某些类Cayley图的全着色问题，验证了全着色猜想，并求得了这些图类的全色数。
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, ....).
A subset $C$ of the vertex set of a graph $Γ$ is called a perfect code of $Γ$ if every vertex of $Γ$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditi…
We introduce a family of graphs that generalises the class of Cayley graphs. For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R) is the directed graph with vertex set G and an arc from x to y if and only if y=a^{-1}xb for some a in L and b in R. Thus, in common with Cayley graphs, two-sided Cayley graphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine when two-sided Cayley graphs are simple undirected graphs, and give sufficient conditions for them to be connected, vertex-transitive, or Cayl…
The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.
Ollivier Ricci curvature of Cayley graphs for dihedral groups, generalized quaternion groups, and cyclic groups
本文基于Lin-Lu-Yau公式计算了二面体群、广义四元数群和循环群在特定小生成集下Cayley图所有边的Ollivier Ricci曲率，给出了完整曲率结果。
In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.
Green showed that there exist constants $C_1,C_2>0$ such that the clique number $ω$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\log n < ω< C_2n\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\mathbb{F}_2^n$ analogue of the famous conjecture by Bollobás and giving almost the complet…
Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytop…
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…
For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
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。
On the degrees of regular nut graphs and Cayley nut graphs
本文证明了对于所有度数 d≥3 均存在无穷多个 d-正则坚果图，并进一步证明对每个偶数 d≥4 存在无穷多个 d-正则凯莱坚果图，从而完全确定了凯莱坚果图的所有可行度数。
A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let Lambda_n be any interval of maximal length in Gamma_n; this graph is the Hasse diagram of the lattice of noncrossing partitions. The meanders of order n are in one-to-one correspondence with ordered pairs of maximally separated vertices of Lambda_n.
A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected Cayley graph on $G$ is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that "many" 2-groups are non-CCA.
A note on Cayley nut graphs whose degree is divisible by four
本文通过构造证明，对所有满足4整除d、d>0、2整除n且n≥d+4的参数，存在d-正则Cayley nut图，从而完全解决了度可被四整除的Cayley nut图与顶点传递nut图的存在性问题。
Non-isomorphic Cayley Graphs with Same Random Walk Distributions
本文构造了一类无限族群及其不同子集，使得对应的凯莱图虽非同构，却具有相同的简单随机游走分布，并揭示了谱集分解中 B1 = -B2 的对称关系。
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。
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。
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。
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