Integral Circulant Graphs Research Articles

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks enabling perfect state transfer. Understanding the potential information transfer between nodes in such networks involves calculating the maximal graph diameter. The integral circulant graph ICGn(D) has vertex set Zn={0,1,2,…,n−1}, with vertices a and b adjacent if gcd⁡(a−b,n)∈D, where D⊆{d:d|n,1≤d<n}. Building on the upper bound 2|D|+1 for the diameter provided by Saxena, Severini, and Shparlinski, we prove that the maximal diameter of ICGn(D) for a given order n with prime factorization p1α1⋯pkαk is r(n) or r(n)+1, where r(n)=k+|{i|αi>1,1≤i≤k}|. We show that a divisor set D with |D|≤k achieves this bound. We calculate the maximal diameter for graphs of order n and divisor set cardinality t≤k, identifying all extremal graphs and improving the previous upper bound.

So's conjecture for integral circulant graphs of 4 types

In [Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of 4 types.

Minimal spread of integral circulant graphs

The integral circulant graph ICGn(D) has the vertex set Zn={0,1,2,…,n−1} and vertices a and b are adjacent if gcd(a−b,n)∈D, where D⊆{d:d∣n,1≤d<n}. In this paper we prove that the minimal value of the spread of the integral circulant graphs ICGn(D) of a given order n with its prime factorization p1α1⋯pkαk, is equal to ∑i=1kpiαi−1 or ∑i=1kpiαi, depending on whether n=2⋅3⋅p3α3⋯pkαk or not, respectively. Furthermore, we completely characterize all graphs whose minimal spread attains the mentioned values in the class of connected integral circulant graphs of a given order.

Characterisation of all integral circulant graphs with multiplicative divisor sets and few eigenvalues

AbstractWe present a method which in principal allows to characterise all integral circulant graphs with multiplicative divisor set having a spectrum, i.e. the set of distinct eigenvalues, of any given size. We shall exemplify the method for spectra of up to four eigenvalues, also reproving some known results for three eigenvalues along the way. In particular we show that given any integral circulant graph of arbitrary order n with multiplicative divisor set and precisely four distinct eigenvalues, n necessarily is either a prime power or the product of two prime powers with explicitly given simply structured divisor set and set of eigenvalues in both cases.

Characterization of strongly regular integral circulant graphs by spectral approach

The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n? 1} and vertices a and b are adjacent if gcd(a ? b, n) ? D, where D ? Dn, Dn = {d : d | n, 1 ? d &lt; n}. Motivated by the incorrect proof of a previously published result, in this paper we characterize the class of integral circulant graphs that are strongly regular. More precisely, connected ICGn(D) is strongly regular if and only if n is composite and D = {d ? Dn | m ? d} for some m | n and n ? 1 ? m ? 2.

Splitting fields of spectra of circulant graphs

Given a circulant graph Cay(Zn,S), we precisely determine its splitting field and algebraic degree, i.e. the least algebraic extension K|Q of the rationals, which contains all eigenvalues of Cay(Zn,S), and its respective degree [K:Q]. This generalizes a result of So who classified all integral circulant graphs. We prove that there is a deep connection between Schur rings and the splitting field of circulant graphs, which are subfields of cyclotomic fields, and solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Moreover, we deduce some new necessary criteria for isospectrality of circulant graphs and give a graph-theoretical interpretation of the algebraic degree of circulant graphs: we prove that the algebraic degree of a circulant graph is determined by its automorphism group.

Quantum state transfer on a class of circulant graphs

ABSTRACT We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous-time quantum walks on quantum networks based on certain circulant graphs with vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize a few previously known results on pretty good state transfer on circulant graphs, and this way we re-discover all integral circulant graphs on vertices exhibiting perfect state transfer.

Integral circulant graphs with four distinct eigenvalues

Let [Formula: see text], the finite cyclic group of order [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. The circulant graph [Formula: see text] is the undirected graph having the vertex set [Formula: see text] and edge set [Formula: see text]. Let [Formula: see text] be a set of positive, proper divisors of the integer [Formula: see text]. In this paper, by using [Formula: see text] we characterize certain connected integral circulant graphs with four distinct eigenvalues.

On the kernel of integral circulant graphs

Given a positive integer n, every integral circulant graph on n vertices is isomorphic to some graph ICG(n,D) having vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd⁡(a−b,n)∈D} for a uniquely determined set D of positive divisors of n. By virtue of its adjacency matrix, one defines the spectrum of a graph G and, naturally, can ask to which degree the eigenvalues of G determine the graph itself. With respect to integral circulant graphs little is known about this question, which is related to a conjecture of So.In this note we examine the role of the eigenvalue 0 and clarify the interrelation between the dimension of the kernel of ICG(n,D) and the graph itself for all prime powers n=pk and for all positive integers n in case D is a multiplicative divisor set.

Structural properties and formulae of the spectra of integral circulant graphs

Structural properties and formulae of the spectra of integral circulant graphs

Pretty Good State Transfer on Circulant Graphs

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS

We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.

Integral circulant Ramanujan graphs via multiplicativity and ultrafriable integers

Any integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd⁡(a−b,n)∈D}. Such graphs are regular, and a connected ρ-regular graph G is called Ramanujan if the second largest modulus of the eigenvalues of the adjacency matrix of G is at most 2ρ−1.In 2010 Droll described all Ramanujan unitary Cayley graphs, i.e. graphs of type Xn:=ICG(n,{1}) having the Ramanujan property. Recently, Le and the author classified all Ramanujan graphs ICG(ps,D) for prime powers ps and arbitrary divisor sets D. We greatly extend the established results to graphs ICG(n,D) with arbitrary n and multiplicative divisor set D: (i) We derive a criterion (in terms of Euler's totient function) for ICG(n,D) to be Ramanujan. (ii) We prove that for all even integers n>2 and a positive proportion of the odd integers n, namely those having a “dominating” prime power factor, there exists a multiplicative divisor set D such that ICG(n,D) is Ramanujan. (iii) We show that the set of odd n for which no Ramanujan graph ICG(n,D) with multiplicative divisor set D exists, viz. ultrafriable integers, has positive density as well. The proofs of (ii) and (iii) use methods from analytic number theory.

On So's conjecture for integral circulant graphs

Each integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z=nZ and edge set {(a,b) : a, b ? Z=nZ, gcd(a - b,n) ? D}. According to a conjecture of So two integral circulant graphs are isomorphic if and only if they are isospectral, i.e. they have the same eigenvalues (counted with multiplicities). We prove a weaker form of this conjecture, namely, that two integral circulant graphs with multiplicative divisor sets are isomorphic if and only if their spectral vectors coincide.

The exact maximal energy of integral circulant graphs with prime power order

The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z},\, \gcd(a-b,n)\in {\cal D}\}$. Given an arbitrary prime power $p^s$, we determine all divisor sets maximising the energy of an integral circulant graph of order $p^s$. This enables us to compute the maximal energy $\Emax{p^s}$ among all integral circulant graphs of order $p^s$.

Which weighted circulant networks have perfect state transfer?

The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini and Shparlinski and characterizing integral graphs among weighted circulant graphs. Finally, classes of weighted circulant graphs supporting perfect state transfer are found. These classes completely cover the class of circulant graphs having perfect state transfer in the unweighted case. In fact, we show that there exists an weighted integral circulant graph with n vertices having perfect state transfer if and only if n is even. Moreover we prove the nonexistence of perfect state transfer for several other classes of weighted integral circulant graphs of even order.

CONVOLUTIONS OF RAMANUJAN SUMS AND INTEGRAL CIRCULANT GRAPHS

There exist several generalizations of the classical Dirichlet convolution, for instance the so-called A-convolutions analyzed by Narkiewicz. We shall connect the concept of A-convolutions satisfying a weak form of regularity and Ramanujan sums with the spectrum of integral circulant graphs. These generalized Cayley graphs, having circulant adjacency matrix and integral eigenvalues, comprise a great amount of arithmetical features. By use of our concept we obtain a multiplicative decomposition of the so-called energy of integral circulant graphs with multiplicative divisor sets. This will be fundamental for the study of open problems, in particular concerning the detection of integral circulant graphs with maximal or minimal energy.

The maximal energy of classes of integral circulant graphs

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a,b}:a,b∈Zn,gcd(a−b,n)∈D}. For a fixed prime power n=ps and a fixed divisor set size |D|=r, we analyse the maximal energy among all matching integral circulant graphs. Let pa1<pa2<⋯<par be the elements of D. It turns out that the differences di=ai+1−ai between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all di equal q:=s−1r−1, or at most the two differences [q] and [q+1] may occur; (ii) there are rules governing the sequence d1,…,dr−1 of consecutive differences. For particular choices of s and r these conditions already guarantee maximal energy and its value can be computed explicitly.

Extremal energies of integral circulant graphs via multiplicativity

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. Integral circulant graphs can be characterised by their order n and a set D of positive divisors of n in such a way that they have vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd(a-b,n)∈D}. Among integral circulant graphs of fixed prime power order ps, those having minimal energy Eminps or maximal energy Emaxps, respectively, are known. We study the energy of integral circulant graphs of arbitrary order n with so-called multiplicative divisor sets. This leads to good bounds for Eminn and Emaxn as well as conjectures concerning the true value of Eminn.

Characterization of quantum circulant networks having perfect state transfer

In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a perfect state transfer (PST) between vertices $${a,b\in V(G)}$$ if |F(?) ab | = 1, for some positive real number ?, where F(t) = exp(i At). Saxena et al. (Int J Quantum Inf 5:417---430, 2007) proved that |F(?) aa | = 1 for some $${a\in V(G)}$$ and $${\tau\in \mathbb {R}^+}$$ if and only if all eigenvalues of G are integer (that is, the graph is integral). The integral circulant graph ICG n (D) has the vertex set Z n = {0, 1, 2, . . . , n ? 1} and vertices a and b are adjacent if $${\gcd(a-b,n)\in D}$$ , where $${D \subseteq \{d : d \mid n, \ 1 \leq d < n\}}$$ . These graphs are highly symmetric and have important applications in chemical graph theory. We show that ICG n (D) has PST if and only if $${n\in 4\mathbb {N}}$$ and $${D=\widetilde{D_3} \cup D_2\cup 2D_2\cup 4D_2\cup \{n/2^a\}}$$ , where $${\widetilde{D_3}=\{d\in D\ |\ n/d\in 8\mathbb {N}\}, D_2= \{d\in D\ |\ n/d\in 8\mathbb {N}+4\}{\setminus}\{n/4\}}$$ and $${a\in\{1,2\}}$$ . We have thus answered the question of complete characterization of perfect state transfer in integral circulant graphs raised in Angeles-Canul et al. (Quantum Inf Comput 10(3&4):0325---0342, 2010). Furthermore, we also calculate perfect quantum communication distance (distance between vertices where PST occurs) and describe the spectra of integral circulant graphs having PST. We conclude by giving a closed form expression calculating the number of integral circulant graphs of a given order having PST.