Large circulant graphs of fixed diameter and arbitrary degree

We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for “stitching” together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset A of a cyclic group of order n such that the k-fold sumset kA is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter

Some identities for enumerators of circulant graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs of prime, twice prime and prime-squared orders. In particular, it is shown that the semi-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes; Cunningham chain; self-complementary graph; tournament; mixed graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4

Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017

Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.Comment: 33 pages, 8 Figure