Related papers: Exponentiation of Graphs

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced…

We propose and study a hierarchical algorithm to generate graphs having a predetermined distribution of cliques, the fully connected subgraphs. The construction mechanism may be either random or incorporate preferential attachment. We…

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…

By generalizing a recent result of Hong, Liu, and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple $k$-node-connected digraphs. More…

In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to…

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such…

Let $G$ be a connected graph with vertex set $\{v_1, v_2, \ldots, v_\mathbf{n}\}$. As a variant of the classical distance matrix, the \emph{exponential distance matrix} was introduced independently by Yan and Yeh, and by Bapat et al. For a…

We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint…

Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their…

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…

A connected graph, on four or more vertices, is matching covered (aka 1-extendable) if every edge is present in some perfect matching. An ear decomposition theorem exists for bipartite matching covered graphs due to Hetyei. From the results…

This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…

We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application,…

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the…

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We…

We study the number of connected graphs with $n$ vertices that cannot be written as the cartesian product of two graphs with fewer vertices. We give an upper bound which implies that for large $n$ almost all graphs are both connected and…

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…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges…