Related papers: Large Cayley graphs of small diameter
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse…
We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost…
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale…
In the present paper, we study Neumaier Cayley graphs. First, we give a criterion for a Cayley graph to be a Neumaier graph with a spread given by the cosets of a subgroup. Further, we construct a new infinite family of Neumaier Cayley…
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…
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…
We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem…
We wish to bring attention to a natural but slightly hidden problem, posed by Erd\H{o}s and Ne\v{s}et\v{r}il in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree…
We classify the distance-regular Cayley graphs with least eigenvalue $-2$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain…
The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…
Let $s$ be a positive integer. Our goal is to find all finite abelian groups $G$ that contain a $2$-subset $A$ for which the undirected Cayley graph $\Gamma(G,A)$ has diameter at most $s$. We provide a complete answer when $G$ is cyclic,…
We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$,…
The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap…
Given a group G, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any…
We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is…
We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.
The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while…
In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…
We show that the total variation mixing time is not quasi-isometry invariant, even for Cayley graphs. Namely, we construct a sequence of pairs of Cayley graphs with maps between them that twist the metric in a bounded way, while the ratio…