A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum computation. In this note, we prove that if a Cayley digraph is a line digraph then it is the pattern of a unitary matrix. We prove that for any finite group with two generators there exists a set of generators such that the Cayley digraph with respect to such a set is a line digraph and hence the pattern of a unitary matrix.
Divisible design digraphs which can be obtained as Cayley digraphs are studied. A characterization of divisible design Cayley digraphs in terms of the generating sets is given. Further, we give several constructions of divisible design Cayley digraphs and classify divisible design Cayley digraphs on $v \le 27$ vertices.
Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.
Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we succeed in finding a method to do so by proving Theorem 1. Also, the number of such Cayley digraphs is computed.
A geometric method for obtaining an infinite family of Cayley digraphs of constant density on finite Abelian groups is presented. The method works for any given degree and it is based on consecutive dilates of a minimum distance diagram associated with a given initial Cayley digraph. The method is used to obtain infinite families of dense or asymptotically dense Cayley digraphs. In particular, for degree $d=3$, an infinite family of maximum known density is proposed.
Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$ we construct Cayley digraphs of order $2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where $k\ge 3$, and bipartite Cayley digraphs of order $2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$, where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge 2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for od…
A digraph X=(V, E) is max-λ, if λ(X)=δ(X). A digraph X is super-λif every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all but a few exceptions, the strongly connected mixed Cayley digraphs are max-λand super-λ.
We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley graph on G that has valence at most 4.
Let X=(V, E) be a digraph. X is maximally connected, if κ(X)=δ(X). X is maximally arc-connected, if λ(X)=δ(X). And X is super arc-connected, if every minimum arc-cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that the strongly connected Bi-Cayley digraphs are maximally connected and maximally arc-connected, and the most of strongly connected Bi-Cayley digraphs are super arc-connected.
Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$ of the concept of congruence in $Z$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known asymptotically dense results were all non-constructive.
We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).
We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.
We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
Given a finite Abelian group $G$ and a generator subset $A\subset G$ of cardinality two, we consider the Cayley digraph $Γ=$Cay$(G,A)$. This digraph is called $2$--Cayley digraph. An extension of $Γ$ is a $2$--Cayley digraph, $Γ'=$Cay$(G',A)$ with $G<G'$, such that there is some subgroup $H<G'$ satisfying the digraph isomorphism Cay$(G'/H,A)\cong$Cay$(G,A)$. We also call the digraph $Γ$ a quotient of $Γ'$. Notice that the generator set does not change. A $2$--Cayley digraph is called optimal when its diameter is optimal with respect to its order. In this work we define two procedures, E and Q,…
Let $Γ=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(Γ)$. The Cayley index of $Γ$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $Γ$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $Γ$ on $G$ whose …
In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as $r$ tends to infinity, for every finite group $R$ of order $r$, out of all possible Cayley digraphs on $R$ the proportion whose automorphism group is as small as possible tends to $1$. This proves a natural conjecture first proposed in $1982$ by Babai and Godsil.
The symmetric group is generated by σ = (1 2 ... n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n.
A finite group $G$ is a "non-DCI group" if there exist subsets $S_1$ and $S_2$ of $G$, such that the associated Cayley digraphs $C\overrightarrow{ay}(G;S_1)$ and $C\overrightarrow{ay}(G;S_2)$ are isomorphic, but no automorphism of $G$ carries $S_1$ to $S_2$. Furthermore, $G$ is a "non-CI group" if the subsets $S_1$ and $S_2$ can be chosen to be closed under inverses, so we have undirected Cayley graphs $Cay(G;S_1)$ and $Cay(G;S_2)$. We show that if $p$ is a prime number, and the elementary abelian $p$-group $(\mathbb{Z}_p)^r$ is a non-DCI group, then $(\mathbb{Z}_p)^{r+3}$ is a non-CI group. I…
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $σ\in Aut(G)$ such that $S^σ=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if…
Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this pap…
Let $Γ=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\not\ni0$. $Γ$ has order ord$(Γ)=|G|=n$ and degree deg$(Γ)=|T|=d$. Let $k(Γ)$ be the diameter of $Γ$ and denote $κ(d,n)=\min\{k(Γ):~\textrm{ord}(Γ)=n,\textrm{deg}(Γ)=d\}$. We give a closed expression, $\ell(d,n)$, of a tight lower bound of $κ(d,n)$ by using the so called {\em solid density} introduced by Fiduccia, Forcade and Zito. A digraph $Γ$ of degree $d$ is called {\em tight} when $k(Γ)=κ(d,|Γ|)=\ell(d,|Γ|)$ holds. Recently, the {\em Dilating Method} has been developed to derive a sequ…
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper n…
We show that for certain integers $n$, the problem of whether or not a Cayley digraph $Γ$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $α\in Aut(G)$ such that $S^α=T$, whenever $Cay(G,S)\cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture …
In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley di…
The support of a matrix M is the (0,1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M. This paper observes some structural properties of digraphs of unitary matrices.
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…
A set $S\subseteq V$ is called an {\em $q^+$-set} ({\em $q^-$-set}, respectively) if $S$ has at least two vertices and, for every $u\in S$, there exists $v\in S, v\neq u$ such that $N^+(u)\cap N^+(v)\neq \emptyset$ ($N^-(u)\cap N^-(v)\neq \emptyset$, respectively). A digraph $D$ is called {\em s-quadrangular} if, for every $q^+$-set $S$, we have $|\cup \{N^+(u)\cap N^+(v): u\neq v, u,v\in S\}|\ge |S|$ and, for every $q^-$-set $S$, we have $|\cup \{N^-(u)\cap N^-(v): u,v\in S)\}\ge |S|$. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for thi…
A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay$(G;T)$, there is a group automorphism $α$ of $G$ such that $S^α=T$. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group $G$ is a CI-group if every Cayley graph on $G$ has the CI property, and is a DCI-group if every Cayley digraph on $G$ has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group, but there are well-known examples of groups that…
In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of G_{n}.
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedr…
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 …
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to some finite subset $S \subseteq \mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$ is the minimum density of a dominating set in $Γ(\mathbb{Z},S)$. We establish some basic results on the domination ratio of $Γ(\mathbb{Z},S)$ and precisely determine it when $S=\{s,t\}$ with $s$ dividing $t$.
The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs from this family are divisible design and integral.
We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{Z}^d$. For any fixed $B$ that generates $\mathbb{Z}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ \cap \mathbb{Z}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.
Dariush Kiani et al.\cite{kiani2015unitary} claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$, but they have only considered two special cases, namely when $n = 2$ and $3$ and have failed to cover the general cases. In this paper, we prove that the unitary Cayley graph of matrix algebras over finite field $F$ is strongly regular if and only if $n=2$.
