Vertices Of Graph Research Articles

PageRank asymptotics on directed preferential attachment networks

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.

Exploration of k-edge-deficient temporal graphs

A temporal graph with lifetime L is a sequence of L graphs G_1, ldots ,G_L, called layers, all of which have the same vertex set V but can have different edge sets. The underlying graph is the graph with vertex set V that contains all the edges that appear in at least one layer. The temporal graph is always connected if each layer is a connected graph, and it is k-edge-deficient if each layer contains all except at most k edges of the underlying graph. For a given start vertex s, a temporal exploration is a temporal walk that starts at s, traverses at most one edge in each layer, and visits all vertices of the temporal graph. We show that always-connected, k-edge-deficient temporal graphs with sufficient lifetime can always be explored in O(kn log n) time steps. We also construct always-connected, k-edge-deficient temporal graphs for which any exploration requires varOmega (n log k) time steps. For always-connected, 1-edge-deficient temporal graphs, we show that O(n) time steps suffice for temporal exploration.

On the Conjecture Related to the Energy of Graphs with Self–Loops

Let Gσ be a graph obtained by attaching a self-loop, or just a loop, for short, at each of σ chosen vertices of a given graph G. Gutman et al. have recently introduced the concept of the energy of graphs with self-loops, and conjectured that the energy E(G) of a graph G of order n is always strictly less than the energy E(Gσ) of a corresponding graph Gσ, for 1 ≤ σ ≤ n − 1. In this paper, a simple set of graphs which disproves this conjecture is exposed, together with some remarks regarding the standard deviations of the (adjacency) eigenvalues of G and Gσ, respectively.

Impacts of Isolated Vertices To Cover Other Vertices in Neutrosophic Graphs

Impacts of Isolated Vertices To Cover Other Vertices in Neutrosophic Graphs

Extremal functions for sparse minors

The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.

Complete Connections Between Vertices in Neutrosophic Graphs

Complete Connections Between Vertices in Neutrosophic Graphs

Graphical designs and gale duality

A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs – cocktail party graphs, cycles, and graphs of hypercubes – by computing or bounding the smallest designs that average all but the last eigenspace in frequency order.

Unique Distance Differentiation By Collection of Vertices in Neutrosophic Graphs

Unique Distance Differentiation By Collection of Vertices in Neutrosophic Graphs

On the hamiltonicity of a planar graph and its vertex‐deleted subgraphs

AbstractTutte proved that every planar 4‐connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex‐deleted subgraphs are hamiltonian. We here prove that if in a planar ‐vertex graph with minimum degree at least 4 at least vertex‐deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every there exists a nonhamiltonian polyhedral ‐vertex graph with minimum degree at least 4 containing hamiltonian vertex‐deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex‐deleted subgraphs as well as Bondy's meta‐conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex‐deleted subgraphs are traceable, must itself be traceable.

Many Turán exponents via subdivisions

AbstractGiven a graph $H$ and a positive integer $n$ , the Turán number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$ -vertex graph that does not contain $H$ as a subgraph. A real number $r\in (1,2)$ is called a Turán exponent if there exists a bipartite graph $H$ such that $\mathrm{ex}(n,H)=\Theta (n^r)$ . A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q\gt p$ .In this paper, we show that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q \gt p^{2}$ . Our result also addresses a conjecture of Janzer [18].

A Theoretical Scientific Programming Framework for Application of Linear Matrix Transformation in Plane Computer Graphics

Plane computer graphics are basic information carriers in many industrial scenarios, such as engineering simulation, automatic control, and software design. Plane computer graphics are generally a kind of digital signals guided by mathematical symbols, and each vertex of a plane computer graph forms a graph matrix. Therefore, linear matrix transformation serves as the most common algorithmic unit to realize various information processing operations. To improve ease of graph matrix computing in practical engineering scenarios, this paper proposes a theoretical scientific programming framework for application of linear matrix transformation in plane computer graphics. Firstly, theoretical basis of linear matrix transformation in homogeneous plane coordinates is displayed and analyzed. Then, the universal theorem about linear transformation of graph matrices is deduced, and corresponding proofs are also given. Finally, a case study is set up to demonstrate the main workflow of the proposed theoretical scientific programming framework. The simulative results reveal feasibility of the proposal.

Multi-agent Pathfinding on Large Maps Using Graph Pruning: This Way or That Way? (Extended Abstract)

This paper extends a study on improving the performance of reduction-based solvers for the problem of multi-agent pathfinding. The task is to navigate a set of agents in a graph without collisions. Solvers that reduce this problem to other formalisms often have issues scaling to larger instances in terms of the graph size. A previous study suggests that pruning the graph of most vertices based on a randomly chosen shortest path for each agent. In this paper, we study the effect of different choices of these paths.

Learning to compute the metric dimension of graphs

For distinct vertices x,y,z in graph G=(V,E), we say the pair of vertices x,y can be distinguished by vertex z, if the distance inequation dG(x,z)≠dG(y,z) is true. If any two different vertices in graph G can be distinguished by at least one vertex in the vertex subset S⊂V then we say S⊂V is a resolving set of graph G. The one with minimum number of nodes among all of the resolving sets is called a metric base of graph G, of which the cardinality is called the metric dimension of graph G. The metric dimension problem(MDP) of computing the metric dimension of graphs is a kind of complex combinatorial optimization problem. In this paper, a hybrid algorithm for solving the MDP is developed, through which the metric base of graphs can be learned, and consequently the metric dimension of graphs is estimated. It is particular that the hybrid algorithm combines graph representation learning with greedy repair policy. Extensive experiments show that the hybrid algorithm has higher computational accuracy and efficiency on random graphs. It is also further suggested that the learning process of MDP is greatly influenced by the attributes of complex networks and the designs of graph neural networks themselves.

Geometric Nature of Relations on Plabic Graphs and Totally Non-negative Grassmannians

Abstract The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicoloured (plabic) graphs in the disc. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations at the vertices of such graphs, depending on some signatures defined on their edges. The problem of characterizing the signatures corresponding to the totally non-negative cells was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non-negativity property on the full positroid cell. If each edge in a graph $\mathcal G$ belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and the gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of the geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property in the case of bipartite graphs [1, 7], and has a different statistical mechanical interpretation otherwise [6]. An explicit connection between the solution of Lam’s system of relations and the value of Postnikov’s boundary measurement map is established using the generalization of Talaska’s formula [51] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for the transformations of the signatures under Postnikov’s moves and reductions and amalgamations of networks.

Triangles in randomly perturbed graphs

AbstractWe study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ . We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$ , where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$ -universality.

Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

Flat origami studies straight line, planar graphs $C=(V,E)$ drawn on a region $R\subset\mathbb{R}^2$ that can act as crease patterns to map, or fold, $R$ into $\mathbb{R}^2$ in a way that is continuous and a piecewise isometry exactly on the faces of $C$. Associated with such crease pattern graphs are valid mountain-valley (MV) assignments $\mu:E\to\{-1,1\}$, indicating which creases can be mountains (convex) or valleys (concave) to allow $R$ to physically fold flat without self-intersecting. In this paper, we initiate the first study of how valid MV assignments of single-vertex crease patterns are related to one another via face-flips, a concept that emerged from applications of origami in engineering and physics, where flipping a face $F$ means switching the MV parity of all creases of $C$ that border $F$. Specifically, we study the origami flip graph $\rm{OFG}(C)$, whose vertices are all valid MV assignments of $C$ and edges connect assignments that differ by only one face flip. We prove that, for the single-vertex crease pattern $A_{2n}$ whose $2n$ sector angles around the vertex are all equal, $\rm{OFG}(A_{2n})$ contains as subgraphs all other origami flip graphs of degree-$2n$ flat origami vertex crease patterns. We also prove that $\rm{OFG}(A_{2n})$ is connected and has diameter $n$ by providing two $O(n^2)$ algorithms to traverse between vertices in the graph, and we enumerate the vertices, edges, and degree sequence of $\rm{OFG}(A_{2n})$. We conclude with open questions on the surprising complexity found in origami flip graphs of this type.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

Average scattering entropy for periodic, aperiodic and random distribution of vertices in simple quantum graphs

This work deals with the average scattering entropy of quantum graphs. We explore this concept in several distinct scenarios that involve periodic, aperiodic and random distribution of vertices of distinct degrees. In particular, we compare distinct situations to see how they behave as we change the arrangements of vertices and the topology and geometry of the proposed structures. The results show that the average scattering entropy may depend on the number of vertices, and on the topological and geometrical disposition of vertices and edges of the quantum graph. In this sense, it can be seen as another tool to be used to explore geometric and topological effects of current interest for quantum systems.

Pewarnaan Titik Ketakteraturan Lokal Refleksif pada Keluarga Graf Tangga

Let a simple and connected graph $G=(V,E)$ with the vertex set $V(G)$ and the edge set E(G). If there is a mapping $f$: $V(G)$ $\rightarrow$ ${0,2,…,2k_v}$ and $f$: $E(G)$ $\rightarrow$ ${1,2,…,k_e}$ as a function of vertex and edge irregularities labeling with $k=max$ ${2k_v,k_e}$ for $k_v$ and $k_e$ natural numbers and the associated weight of vertex $u,v \in V(G)$ under $f$ is $w(u)=f(u)+\sum_{u,v\in E(G)}f(uv)$. Then the function $f$ is called a local vertex irregular reflexive labeling if every adjacent vertices has distinct vertex weight. When each vertex of graph $G$ is colored with a vertex weight $w(u,v)$, then graph $G$ is said to have a local vertex irregular reflexive coloring. Minimum number of vertex weight is needed to color the vertices in graf $G$ such that any adjacent vertices are not have the same color is called a local vertex irregular reflexive chromatic number, denoted by $\chi_{(lrvs)}(G)$. The minimum $k$ required such that $\chi_{(lrvs)}(G)=\chi(G)$ where $\chi(G)$ is chromatic number of proper coloring on G is called local reflexive vertex color strength, denoted by $lrvcs(G)$. In this paper, we will examine the local reflexive vertex color strength of local vertex irregular reflexive coloring on the family of ladder graph.

A Link Prediction Algorithm Based on GAN

Link prediction, as an important research direction in complicated network analysis, has broad application prospects. However, traditional link prediction algorithms are generally designed by the sparse expression of the adjacency matrix, which is computationally expensive and inefficient, being also unable to run on large-scale networks and to preserve their higher order structural features. To fill this gap, we propose a GAN (generative adversarial network)-based link prediction algorithm. The algorithm layers the network graph, preserving the local features and higher-level structural features of the original network graph, and uses a generative adversarial model to recursively and backwardly obtain the low-dimensional vector form of the vertices in each layer of the network graph as the initialization of the network graph in the previous layer. It then obtains the low-dimensional vector form of all the vertices in the original network graph for link prediction, and the problem of local minima that can be generated by random initialization is solved. The experimental results show that our method is superior to many state-of-the-art algorithms.