Related papers: Embedding of metric graphs on hyperbolic surfaces
A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded…
We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result. More precisely, it is shown that the algebraic connectivity of a surface S, defined as the supremum of a(G) over all…
The maximum genus $\gamma_M(G)$ of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal…
This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal…
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general…
The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C…
Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces…
A {\em rooted graph} is a graph together with a designated vertex subset, called the {\em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {\em face cover} if every…
Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical…
In this work, we study various properties of embedded hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^{\mu}$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper…
While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually…
Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots,…
In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular…
Let $G$ be a plane bipartite graph and $\mathcal{M}(G)$ be the set of all perfect matchings of $G$. The resonance graph $R(G)$ is a graph whose vertex set is $\mathcal{M}(G)$, and two perfect matchings are adjacent in $R(G)$ if their…
Embedding graphs in continous spaces is a key factor in designing and developing algorithms for automatic information extraction to be applied in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings…
In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph $G$, we say that a plane subgraph $H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge (possibly with…
Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the…
Negami's famous planar cover conjecture is equivalent to the statement that a connected graph can be embedded in the projective plane if and only if it has a projective planar cover. In 1999, Hlin\v{e}n\'y proposed extending this conjecture…
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface…
The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean…