Related papers: Undirected Graphs of Entanglement 3
A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply…
Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex.…
Every end of an infinite graph $ G $ defines a tangle of infinite order in $ G $. These tangles indicate a highly cohesive substructure in the graph if and only if they are closed in some natural topology. We characterize, for every finite…
In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of…
We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and…
A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for…
The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of…
Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a…
We investigate a combinatorial reconfiguration problem on oriented graphs, where a reconfiguration step (edge-flip) is the inversion of the orientation of a single edge. A recently published conjecture that is relevant to the correctness of…
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a…
A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a…
Many optimization, inference and learning tasks can be accomplished efficiently by means of decentralized processing algorithms where the network topology (i.e., the graph) plays a critical role in enabling the interactions among…
We introduces the umodules, a generalisation of the notion of graph module. The theory we develop captures among others undirected graphs, tournaments, digraphs, and $2-$structures. We show that, under some axioms, a unique decomposition…
An (edge) decomposition of a graph $G$ is a set of subgraphs of $G$ whose edge sets partition the edge set of $G$. Here we show, for each odd $\ell \geq 5$, that any graph $G$ of sufficiently large order $n$ with minimum degree at least…
In this paper, we introduce the notion of a finite non-simple directed graph, called an ornated graph and initiate a study on ornated graphs. An ornated graph is a directed graph on $n$ vertices, denoted by $O_n(s_l)$, whose vertices are…
Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we…
We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of…
We propose a network protocol similar to the $k$-tree protocol of Itai and Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we define an {\em $t$-uncovering-by-bases} for a connected graph $G$ to be a collection…
The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a…
This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called a Base Graph -- Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence…