Extremal energies of integral circulant graphs via multiplicativity

AbstractThe 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

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation \pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).Comment: 11 pages, no figure

Counting sets with small sumset, and the clique number of random Cayley graphs

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). Despite the graph-theoretical title, this is a paper in number theory. Our main results are essentially estimates for the number of sets A in {1,...,N} with |A| = k and |A + A| = m, for various values of k and m.Comment: 18 pages; to appear in Combinatorica, exposition has been improved thanks to comments from Imre Ruzsa and Seva Le

A quadratic lower bound for subset sums

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.Comment: 12 page

Computing by Temporal Order: Asynchronous Cellular Automata

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity). Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249