arXiv research

The study proves unique harmonic functions and combinatorial properties of vertex-transitive graphs.

This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.

An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph.

This paper shows semi-equivelar toroidal maps are vertex-transitive covers.

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particu…

We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of act…

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse…

634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.

e-GGPs learn graph vertex transitions over time.

A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …

Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type $(3^5, 4)$ on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on…

If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the torus is a quotient of an Archimedean tiling on the plane then the map…

We enumerate and classify all the semi equivelar maps on the surface of $χ=-2$ with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.

Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cy…

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polyt…

Groups acting on product trees are boundary rigid.

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $β^k$ into closed surfaces of genus $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-action on $β^k$. Furthermore we show that for each $k \equiv …

We present a constructive proof, that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope β^k into closed surfaces of genus \leq 1, each with a transitive automorphism group given by the vertex transitive Z_{2k}-action on β^k. Furthermore we show, that for each k \equiv 1,5(6) the 2-skele…

Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than $2-$Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangula…

We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated …

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparam…

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…

The Wythoff construction takes a $d$-dimensional polytope $P$, a subset $S$ of $\{0,..., d\}$ and returns another $d$-dimensional polytope $P(S)$. If $P$ is a regular polytope, then $P(S)$ is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want …

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…

Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.

Proposes MGMN for end-to-end graph similarity learning.

The paper explores graphons of line graphs from sparse finite graphs.

MxPool learns graph features from diverse graphs using a hierarchical structure.

Study the geometry of graph product extension graphs.

Graph neural network learns graph distances effectively.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

Customized-GNN generates model-specific for each graph.

Graph embedding leaks sensitive graph properties and subgraphs.

Characterizes graphs with leveled embeddings and introduces new graph invariants.

The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.

Two new methods improve graph embedding without needing a complete graph structure.

We define a pseudo-inverse for line graphs using linear integer programming.

Develops method to create non-Abelian Ricci-flat graphs via bundles.

New method uses graph generative models for graph classification.

MathNet uses wavelets for graph representation and learning.

Unified framework for graph coarsening using node features and graph matrices.

A fast graph embedding method for large graphs.

Quadratic bounds found for graph dimensions.

Study classifies Halin graphs with positive curvature.

PSimGNN partitions graphs into subgraphs for efficient graph similarity computation.

We present graph wavelet neural network (GWNN), a novel graph convolutional neural network (CNN), leveraging graph wavelet transform to address the shortcomings of previous spectral graph CNN methods that depend on graph Fourier transform. Different from graph Fourier transform, graph wavelet transform can be obtained …

This paper develops graph theory for racks and quasigroups.