Fan Wu
In this paper, generalizing the result in \cite{GXY}, we construct strongly regular Cayley graphs by using union of cyclotomic classes of $\F_q$ and Gauss sums of index $w$, where $w\geq 2$ is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.
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We give a construction of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.
Unimodular covers of 3-dimensional parallelepipeds and Cayley sums
本文证明了三维空间中平行六面体、中心对称多胞体及特定Cayley和类具有幺模覆盖，改进了先前IDP结果。
Cayley sums and Minkowski sums of lattice polytopes
本文以 Minkowski 和刻画 Cayley 和的整数分解性质，并据此研究 2-凸正规格多面体两类和的该性质，同时讨论其 level 性质。
In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [24] and several of subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.
If X is any connected Cayley graph on any finite abelian group, we determine precisely which flows on X can be written as a sum of hamiltonian cycles. (This answers a question of Brian Alspach.) In particular, if the degree of X is at least 5, and X has an even number of vertices, then the flows that can be so written are precisely the even flows, that is, the flows f, such that the sum of the edge-flows of f is divisible by 2. On the other hand, there are examples of degree 4 in which not all even flows can be written as a sum of hamiltonian cycles. Analogous results were already known, from …
Random Cayley graphs and random sumsets
该研究证明有限阿贝尔群存在特殊结构子集族，由此改进了稀疏随机凯莱图等的独立数、非和集最小规模的界，并得到稀疏随机子集和集算术级数最大长度的紧界。
A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.
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…
In this paper, we give a construction of strongly regular Cayley graphs on the additive groups of finite fields based on three-valued Gauss periods. As consequences, we obtain two infinite families and one sporadic example of new strongly regular Cayley graphs. This construction can be viewed as a generalization of that of strongly regular Cayley graphs obtained in \cite{BLMX}.
Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modification of his arguments shows that for an arbitrary finite abelian group of order $n$, there is a Cayley graph containing…
Given a set A in Z/NZ we may form a Cayley sum graph G_A on vertex set Z/NZ by joining i to j if and only if i + j is in A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G_A is a.s. O(log N). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on Z/NZ. Indeed, we also show that the clique number of a random Cayley sum graph on (Z/2Z)^n, 2^n = N, is almost surely not O(log N). Despi…
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
In this paper, we study new Cayley graphs over the additive group of Galois rings. First we prove that they are expander graphs by using a Weil-Carlitz-Uchiyama type estimation of character sums for Galois rings. We also show that Cayley graphs from Galois rings of characteristic 4 form a new infinite family of Ramanujan graphs by an elementary eigenvalue estimation. Moreover some other spectral properties of our graphs are also discussed.
Hamilton cycles in generalized dihedral Cayley graphs and digraphs
本文证明了广义二面体群上连通Cayley有向图存在有向Hamilton圈，证实了1978年的猜想，关键新工具是三次Haar图的三重和集覆盖定理。
Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization
该文针对二面体群上Cayley图的完美态转移问题，通过数学方法完成完整实用表征，为识别构建此类图及确定最小转移时间提供准则。
If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers}, then the mixed graph is called \emph{Gaussian integral}. For any finite group $Γ$, we characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if it is Gaussian integral.
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。
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.
Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^α=T$ for some automorphism $α$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_ν^S=M_ν^T$ for all positive integers $ν$, where $M_ν…
Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012) studied Gauss sums, some integral powers of which are in the field of rational numbers. Such Gauss sums are called {\it pure}. In particular, Aoki (2004) gave a necessary and sufficient condition for a Gauss sum to be pure in terms of Dirichlet characters modulo the order of the multiplicative character involved. In this paper, we study pure Gauss sums with odd extension degree $f$ and classify them for $f=5,7,9,11,13,17,19,23$ based on Aoki's theorem. Furthermore, we characterize a special subclass of pure Gauss su…
Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.
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。
The duals of cyclic codes with two zeros have been extensively studied, and their weight distributions have recently been evaluated in some cases. In this note, we determine the weight distribution of a certain new class of such codes by computations involving index 2 Gauss sums.
We give two constructions of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.
In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters $(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$, where $r=M(q^2-1)/2$, in the following cases: (i) $M=1$ and $q\equiv 3\,(\mod{4})$; (ii) $M=3$ and $q\equiv 7\,(\mod{24})$; and (iii) $M=7$ and $q\equiv 11,51\,(\mod{56})$. The existence of strongly regular Cayley graphs with the above para…
Butson Hadamard Matrices from Pairs of Gauss Sums
本文提出一种利用两个高斯和构造 Butson Hadamard 矩阵的新方法，适用于素数幂 $q$ 及 $q-1$ 的任意因子 $d$，并通过 Butson-Jacobsthal 函数推广了 Jacobsthal 矩阵构造。
In the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the order $m$ of the multiplicative character involved. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields. On the other hand, Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004…
In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds by Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ].
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。
Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of $GP(q,d)$ is tight if and only if $d \mid (\sqrt{q}+1)$, which strengthens the previous related results by Broere-Döman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickel…
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that, except for a few cases with small $d$, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ra…
We use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix rings to show that every $n \times n$ nonzero matrix over $\Bbb F_q$ can be written as a sum of two $\operatorname{SL}_n$-matrices when $n>1$. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if $X$ is a subset of $\operatorname{Mat}_2 (\Bbb F_q)$ with size $|X| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $X$ contains at least two distinct matrices whose difference has determinant $α$ for any $α\in \Bbb F_q^{\ast}$. Using this result we also prove a …
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