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 expander families of bounded degree, whose spectra are related by the values of certain characters. We also provide several new examples of expander families, and examples of non-expanders and Ramanujan graphs formed by these three variants.
Let $G$ be a finite group with $|G|\geq 4$ and $S$ be a subset of $G$. Given an automorphism $σ$ of $G$, the twisted Cayley graph $C(G, S)^σ$ (resp. the twisted Cayley sum graph $C_Σ(G, S)^σ$) is defined as the graph having $G$ as its set of vertices and the adjacent vertices of a vertex $g\in G$ are of the form $σ(gs)$ (resp. $σ(g^{-1} s)$) for some $s\in S$. If the twisted Cayley graph $C(G, S)^σ$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree, the order of $σ$ an…
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.
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.
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 determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{λ,-λ\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum grap…
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 …
We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$ large enough.
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 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.
A perfect code $C$ in a graph $Γ$ is an independent set of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $Γ$ is a set of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of…
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.
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.
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…
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_Σ(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}η, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_Σ(G,S)$ and $η= 2…
It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set…
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…
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…
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…
In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of the codes in this construction are also perfect. The graphs considered include some well-known graphs such as tori, twisted tori and Kronecker products of two cycles. Therefore, the codes can be used for identification in these graphs. Finally, an example of how these codes can be applied for adaptive identification over these graphs is presented.
In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform $2$-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in th…
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…
A signed dominating function of graph $Γ$ is a function $g :V(Γ) \longrightarrow \{-1,1\}$ such that $\sum_{u \in N[v]}g(u) >0$ for each $v \in V(Γ)$. The signed domination number $γ_{_S}(Γ)$ is the minimum weight of a signed dominating function on $Γ$. Let $G=\langle S \rangle$ be a finite group such that $e \not\in S=S^{-1}$. In this paper, we obtain the signed domination number of $Cay(S:G)$ based on cardinality of $S$. Also we determine the classification of group $G$ by $|S|$ and $γ_{_S}(Cay(S:G))$.
In this paper, we determine an explicit formula for the number of walks in $X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, the unitary Cayley Graphs of order $n$, between any pair of its vertices. With this result, we give the number of representations of a fixed residue class $\bmod{}n$ as the sum of $k$ units of $\mathbb{Z}_n$.
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 …
Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $Φ_{k+r}(n_1,n_2,...,n_r)$, where $Φ_{\ell}(n_1,n_2,...,n_r)= \sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}} \prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\ge{}4$, $2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for a…
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.
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.
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…
We show that certain canonical realizations of the complexes Hom(G,H) and Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of such projected Hom-complexes: the dissections of a convex polygon into k-gons, Postnikov's generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, a…
Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $α\in [0,1]$, and any $S \subseteq G$ with $|S| = \frac{|G|}{q+α}$, we show $\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\max\left(\frac{q^2-αq+α^2}{q^2},\frac{q^2+2αq+4α^2-6α+3}{(q+1)^2},γ_0\right)$, where $γ_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, and Sudakov for Cayley graphs.
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 …
We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix…
