Related papers: Ramanujan Graphs with Small Girth
We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish…
Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.
In this paper we present a novel framework for sequence to shape maps. These combinatorial maps realize exponentially many shapes, and have preimages which contain extended connected subgraphs of diameter n (neutral networks). We prove that…
The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which…
High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are…
We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total…
Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application…
In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1, G_2, G_3, \ldots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that…
The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded…
We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The…
This paper presents a characterization of edge-transitive graphs which are four regular and have girth four. This class consists of four infinite families plus four exceptional graphs.
We construct infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for many higher rank parabolic model geometries. To do this, we use particular elements of the…
We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as $2^n-$rational maps. In this note we construct a rich family of $2^n-$rational maps. These…
Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $\Gamma_n$. We construct self-similar groups, whose graphs $\Gamma_n$ can be represented as an iterated zig-zag product and…
We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…
We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…
This is an item on Ramanujan Graphs for a planned encyclopedia on Ramanujan. The notion of Ramanujan graphs is explained, as well as the reason to name these graphs after Ramanujan.
We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a `small' covering set. In particular, we give bounds on the…
In the generalized truncation construction, one replaces each vertex of a $k$-regular graph $\Gamma$ with a copy of a graph $\Upsilon$ of order $k$. We investigate the symmetry properties of the graphs constructed in this way, especially in…
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such…