Related papers: Minimal Charts

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical…

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…

Let $\Gamma$ be a chart, and we denote by $\Gamma_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(7)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=7$ where $w(G)$ is the number…

We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…

We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives…

We consider the task of drawing a graph on multiple horizontal layers, where each node is assigned a layer, and each edge connects nodes of different layers. Known algorithms determine the orders of nodes on each layer to minimize crossings…

The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…

This paper presents a complete classification of minimal graph surfaces that admit graphical transformations into other minimal surfaces. These transformations are functions that map the height function of a minimal graph surface to another…

In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.

In this article we investigate the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group. After proving existence of (possibly degenerate) minimal horizontal triods, we…

Trace diagrams are structured graphs with edges labeled by matrices. Each diagram has an interpretation as a particular multilinear function. We provide a rigorous combinatorial definition of these diagrams using a notion of signed graph…

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…

Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list…

After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their…

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…

A minimal family of curves on an embedded surface is defined as a 1-dimensional family of rational curves of minimal degree, which cover the surface. We classify such minimal families using constructive methods. This allows us to compute…

We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph…

We study braid diagrams with a minimal number of crossings. Such braid diagrams correspond to geodesic words for the braid groups with standard Artin generators. We prove that a diagram of a homogeneous braid is minimal if and only if it is…