Related papers: Regular maps of high density

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency…

The pentagram map is a discrete dynamical system on planar polygons. By definition, the image of a polygon $P$ under the pentagram map is the polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$.…

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of…

Given two regular graphs with consistent rotation maps, we produce a constructive method for a consistent rotation map on their Cartesian product. This method will be given as a simple set of rules of addition and table look ups. We assume…

A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.

A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 vertices are known only for $n \in \{52, 54, 57, 60\}$, we prove…

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable…

We reconsider the geometry of pure and mixed states in a finite quantum system. The rangesof eigenvalues of the density matrices delimit a regular simplex (Hypertetrahedron TN) in any dimension N; the polytope isometry group is the…

A hypergraph is linear if each pair of distinct vertices appears in at most one common edge. We say $\varGamma=(V,E)$ is an associated graph of a linear hypergraph $\mathcal{H}=(V, X)$ if for any $x\in X$, the induced subgraph…

We classify the regular maps $\mathcal M$ which have automorphism groups $G$ acting faithfully and primitively on their vertices. As a permutation group $G$ must be of almost simple or affine type, with dihedral point stabilisers. We show…

Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified. A nature question is to classify regular Cayley maps of elementary abelian $p$-groups $Z_p^n$. In this paper, a complete classification of regular…

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists…

Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for…

It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional…

A new family of strongly regular graphs, called the general symplectic graphs $Sp(2\nu, q)$, associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their…

The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…

A stable map of a closed orientable $3$-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links…

A nearly platonic graph is a k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for a finite graph to have exactly one disparate face, and offer some conjectures,…