Related papers: Higher dimensional digraphs from cube complexes an…
A graph $G$ is said to be a $k$-degenerate graph if any subgraph of $G$ contains a vertex of degree at most $k$. Let $\kappa$ be any non-negative function on the vertex set of $G$. We first define a $\kappa$-degenerate graph. Next we give…
Double triangle expansion is an operation on $4$-regular graphs with at least one triangle which replaces a triangle with two triangles in a particular way. We study the class of graphs which can be obtained by repeated double triangle…
Tree-width is an invaluable tool for computational problems on graphs. But often one would like to compute on other kinds of objects (e.g. decorated graphs or even algebraic structures) where there is no known tree-width analogue. Here we…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the…
Let $G$ be a topological group acting on a simplicial complex $\mathcal{X}$ satisfying some mild assumptions. For example, consider a $k$-regular tree and its automorphism group, or more generally, a regular affine Bruhat-Tits building and…
A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a…
A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na…
We address the problem of representing dynamic graphs using $k^2$-trees. The $k^2$-tree data structure is one of the succinct data structures proposed for representing static graphs, and binary relations in general. It relies on compact…
The paper introduces the notion of a representation $k$-graph $(\Delta,\alpha)$ for a given $k$-graph $\Lambda$. It is shown that any representation $k$-graph for $\Lambda$ yields a module for the Kumjian-Pask algebra $KP(\Lambda)$, and the…
Hypergraphs require higher-dimensional representations, which makes it more difficult to compute and interpret their spectral properties. This survey article uses the framework of hypermatrices to give an in-depth overview of the spectral…
In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded degree…
Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these…
Let $G$ be a connected graph. For an ordered set $S=\{v_1,\ldots, v_\ell\}\subseteq V(G)$, the vector $r_G(v|S) = (d_G(v_1,v), \ldots, d_G(v_\ell,v))$ is called the metric $S$-representation of $v$. If for any pair of different vertices…
A $k$-dispersed labelling of a graph $G$ on $n$ vertices is a labelling of the vertices of $G$ by the integers $1, \dots , n$ such that $d(i,i+1) \geq k$ for $1 \leq i \leq n-1$. $DL(G)$ denotes the maximum value of $k$ such that $G$ has a…
A cut in a digraph $D=(V,A)$ is a set of arcs $\{uv \in A: u\in U, v\notin U\}$, for some $U\subseteq V$. It is known that the arc set $A$ is covered by $k$ cuts if and only if it admits a $k$-coloring such that no two consecutive arcs $uv,…
We generalize the concept of token graphs to obtain supertoken graphs. In the latter case, there can be more than one token in a vertex. We formally define supertoken graphs and establish their basic properties. Moreover, we provide some…
Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $\Pi(n,k)_q$ formed…
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A…
Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This…