Special embeddings of Cayley diagrams

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K̃,L̃ in orientable surfaces S̃, such that S̃ are branched covers of S and the restrictions of the branched coverings to K̃,L̃ are wqc's of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK1] from embeddings of “quotient” graphs G.

Automorphisms of Riemann surfaces

Classification of nonorientable regular embeddings of complete bipartite graphs

Graphs are commonly represented as points in space connected by lines. The points in space are the vertices of the graph, and the lines joining them are the edges of the graph. A general definition of a graph is considered here, where multiple edges are allowed between two vertices and an edge is permitted to connect a vertex to itself. It is assumed that graphs are connected, i.e., any vertex in the graph is reachable from another distinct vertex either directly through an edge connecting them or by a path consisting of intermediate vertices and connecting edges. Under this visual representation, graphs can be drawn on various surfaces. The focus of my research is restricted to a class of surfaces that are characterized as compact connected orientable 2-manifolds. The drawings of graphs on surfaces that are of primary interest follow certain prescribed rules. These are called 2-cellular graph embeddings, or simply embeddings. A well-known closed formula makes it easy to enumerate the total number of 2-cellular embeddings for a given graph over all surfaces. A much harder task is to give a surface-wise breakdown of this number as a sequence of numbers that count the number of 2-cellular embeddings of a graph for each orientable surface. This sequence of numbers for a graph is known as the genus distribution of a graph. Prior research on genus distributions of graphs has primarily focused on making calculations of genus distributions for specific families of graphs. These families of graphs have often been contrived, and the methods used for finding their genus distributions have not been general enough to extend to other graph families. The research I have undertaken aims at developing and using a general method that frames the problem of calculating genus distributions of large graphs in terms of a partitioning of the genus distributions of smaller graphs. To this end, I use various operations such as edge-amalgamation , self-edge-amalgamation, and vertex-amalgamation to construct large graphs out of smaller graphs, by coupling their vertices and edges together in certain consistent ways. This method assumes that the partitioned genus distribution of the smaller graphs is known or is easily calculable by computer, for instance, by using the famous Heffter-Edmonds algorithm. As an outcome of the techniques used, I obtain general recurrences and closed-formulas that give genus distributions for infinitely many recursively specifiable graph families. I also give an easily understood method for finding non-trivial examples of distinct graphs having the same genus distribution. In addition to this, I describe an algorithm that computes the genus distributions for a family of graphs known as the 4-regular outerplanar graphs.

Regular maps from Cayley graphs II antibalanced Cayley maps

We prove that, with the single exception of the 2‐group C, the Cayley table of each Abelian group appears in a face 2‐colorable triangular embedding of a complete regular tripartite graph in an orientable surface. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 71–83, 2010

In this work we will find the minimun genus of a compact non orientable Riemann Surface, having a. given. Cyclic group of . automorphisms . For this we caracterize the proper N. E. C . groups for which there is a non-orientable Surface-Kernel homomorphism . In the last part of this paper we give a bound depending on: the genus, for the order of the cyclic groups of automorphisms of compact Klein Surfaces without boundary, and cyclic groups of automorphisms reaching the bound are obtained.

The classical theorem of Fary states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fary’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil–Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

Let $G$ be a graph embedded in an orientable surface $\Sigma$, possibly with edge weights, and denote by ${\rm len}(\gamma)$ the length (the number of edges or the sum of the edge weights) of a cycle $\gamma$ in $G$. The stretch of a graph embedded on a surface is the minimum of ${\rm len}(\alpha)\cdot {\rm len}(\beta)$ over all pairs of cycles $\alpha$ and $\beta$ that cross exactly once. We provide two algorithms to compute the stretch of an embedded graph, each based on a different principle. The first algorithm is based on surgery and computes the stretch in time $O(g^4 n \log n)$ with high probability, or in time $O(g^4 n \log^2 n)$ in the worst case, where $g$ is the genus of the surface $\Sigma$ and $n$ is the number of vertices in $G$. The second algorithm is based on using a short homology basis and computes the stretch in time $O(n^2\log n + n^2g + ng^3)$.

Classification of reflexible Cayley maps for dihedral groups

Surface segmentation is a fundamental problem in computer graphics.\nIt has various applications such as metamorphosis, surface matching,\nsurface compression, 3D shape retrieval, texture mapping, etc. All\norientable surfaces are Riemann surfaces, and admit conformal structures.\nThis paper introduces a novel surface segmentation algorithm based on\nits conformal structure. Each segment can be conformally mapped to a\nplanar rectangle, and the transition maps are planar translations.\nThe segmentation is intrinsic to the surface, independent of the embedding,\nand consistent for surfaces with similar geometries. By using segmentation\nbased on conformal structure, the mapping between surfaces with arbitrary\ntopologies can be constructed explicitly. The method is rigorous, efficient\nand automatic. The segmentation can be applied to surface morphing,\nconstruct conformal geometry image, convert mesh to Spline surface, solve\nPartial Differential Equations on arbitrary surfaces, etc.

Emerging technologies for drug repurposing: Harnessing the potential of text and graph embedding approaches

ExEm: Expert embedding using dominating set theory with deep learning approaches

Inferring social relationships from user location data has become increasingly important for real-world applications, such as recommendation, advertisement targeting, and transportation scheduling. Most existing mobility relationship measures are based on pairwise meeting frequency, that it, the more frequently two users meet (i.e., co-locate at the same time), the more likely that they are friends. However, such frequency-based methods suffer greatly from data sparsity challenge. Due to data collection limitation and bias in the real world (e.g., check-in data), the observed meeting events between two users might be very few. On the other hand, existing methods focus too much on the interactions between two users, but fail to incorporate the whole social network structure. For example, the relationship propagation is not well utilized in existing methods. In this paper, we propose to construct a user graph based on their spatial-temporal interactions and employ graph embedding technique to learn user representations from such a graph. The similarity measure of such representations can well describe mobility relationship and it is particularly useful to describe the similarity for user pairs with low or even zero meeting frequency. Furthermore, we introduce semantic information on meeting events by using point-of-interest (POI) categorical information. Additionally, when part of the social graph is available as friendship ground truth, we can easily encode such online social network information through a joint graph embedding. Experiments on two real-world datasets demonstrate the effectiveness of our proposed method.

Entity Resolution is the process of identifying whether or not various entities from different sources are referring to the same real-world entity. Entity Resolution hasn't been extensively researched in graph databases, whereas it has been for relational databases. This paper focuses on providing comparisons of experiments on various datasets to determine the most appropriate method used in the Entity Resolution process from among literature's similarity algorithms, graph embedding techniques, and graph embedding algorithms combined to link prediction. Moreover, if the embedding algorithm employed has an impact on the given results. The results show that the Entity Resolution process performed better when graph embedding techniques were paired with link prediction, and the chosen graph embedding algorithm also has an impact on the results.