Vertices Of Graph Research Articles

Automatically Testing Containedness between Geometric Graph Classes defined by Inclusion, Exclusion, and Transfer Axioms under Simple Transformations

We study classes of geometric graphs, which all correspond to the following structural characteristic. For each instance of a vertex set drawn from a universe of possible vertices, each pair of vertices is either required to be connected, forbidden to be connected, or existence or non-existence of an edge is undetermined. The conditions which require or forbid edges are universally quantified predicates defined over the vertex pair, and optionally over existence or non-existence of another edge originating at the vertex pair. We consider further a set of simple graph transformations, where the existence of an edge between two vertices is logically determined by the existence or non-existence of directed edges between both vertices in the original graph. We derive and prove the correctness of a logical expression, which is a necessary and sufficient condition for containedness relations between graph classes that are described this way. We apply the expression on classes of geometric graphs, which are used as theoretical wireless network graph models. The models are constructed from three base class types and intersection combinations of them, with some considered directly and some considered as symmetrized variants using two of the simple graph transformations. Our study then goes systematically over all possible graph classes resulting from those base classes and all possible simple graph transformations. We derive automatically containedness relations between those graph classes. Moreover, in those cases where containedness does not hold, we provide automatically derived counter examples.

On neighborhood inverse sum indeg index of molecular graphs with chemical significance

Chemical graph theory is an interdisciplinary field that analyses the molecular structure of a chemical compound as a graph and investigates related mathematical queries by employing graph theoretical and computational techniques. The topological index is an important tool in this area that associates a numerical value with a graph structure. It can be interpreted as a real-valued function that represents the physico-chemical information of a chemical compound. One of the most recently reported neighbourhood degree sum-based indices is the neighbourhood inverse sum indeg index (NI). In this work, we first investigate the application potential of the NI index by exploring its predictive potential and isomer discrimination ability. Following that, some fascinating mathematical features of NI are revealed. Extremal values of NI are estimated for the class of all trees and unicyclic graphs. Furthermore, some crucial upper bounds on NI are set up in terms of well-known graph parameters, including graph order,independence number, and vertex connectivity. Associations between NI and existing indices are also demonstrated.

Classifying the globally rigid edge‐transitive graphs and distance‐regular graphs in the plane

AbstractA graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge‐length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex‐transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex‐transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge‐transitive graphs and distance‐regular graphs are globally rigid by their minimal and maximal degrees.

Investigating the Diversity of Tuberculosis Spoligotypes with Dimensionality Reduction and Graph Theory

The spoligotype is a graphical description of the CRISPR locus present in Mycobacterium tuberculosis, which has the particularity of having only 68 possible spacers. This spoligotype, which can be easily obtained either in vitro or in silico, allows to have a summary information of lineage or even antibiotic resistance (when known to be associated to a particular cluster) at a lower cost. The objective of this article is to show that this representation is richer than it seems, and that it is under-exploited until now. We first recall an original way to represent these spoligotypes as points in the plane, allowing to highlight possible sub-lineages, particularities in the animal strains, etc. This graphical representation shows clusters and a skeleton in the form of a graph, which led us to see these spoligotypes as vertices of an unconnected directed graph. In this paper, we therefore propose to exploit in detail the description of the variety of spoligotypes using a graph, and we show to what extent such a description can be informative.

A bipartite version of the Erdős–McKay conjecture

AbstractAn old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$ -vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ . We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$ , then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ .

Improving node embedding by a compact neighborhood representation

Graph Embedding, a learning paradigm that represents graph vertices, edges, and other semantic information about a graph into low-dimensional vectors, has found wide applications in different machine learning tasks. In the past few years, we have had a plethora of methods centered on graph embedding using different techniques, such as spectral classification, matrix factorization and learning. In this context, choosing the appropriate dimension of the obtained embedding remains a fundamental issue. In this paper, we propose a compact representation of a node’s neighborhood, including attributes and structure, that can be used as an additional dimension to enrich node embedding, to ensure accuracy. This compact representation ensures that both semantic and structural properties of a node’s neighboring-hood are properly captured in a single dimension. Consequently, we improve the learned embedding from state-of-the-art models by introducing the neighborhood compact representation for each node as an additional layer of dimensionality. We leverage on this neighborhood encoding technique and compare with embedding from state-of-the-art models on two learning tasks: node classification and link prediction. The performance evaluation shows that our approach gives a better prediction and classification accuracy in both tasks.

Chromatic Number of Amalgamation of Wheel Graph-Star Graph and Amalgamation of Wheel Graph-Sikel Graph

<p>A graph denoted by is a pair of where is a non-empty set of vertices in G, and E is a set of edges in G. In graph theory, there are various types of graphs including star graphs, cycle graph, and wheel graph. Graph operations on two or more types of graphs can produce new graphs. Amalgamation is one of the operations on graphs. Suppose and are two connected graphs, amalgamating the vertices of graph 1 and graph 2 by joining vertices and vertices denoted by is the graph obtained by joins vertex of graph and vertex of graph into one vertex , where is the common vertex of the graph resulting from , while amalgamating the edges of graph and graph by joining edges and edge , denoted by are graphs obtained by combining edge of graph 1 and edge of graph 2 into one edge g, where g is common side of graph of (𝐺1,𝐺2;𝑒,𝑓). Vertex coloring is one of the fastest-growing topics in graphs. The coloring of the vertices, that is, assigning a color to each vertex of the graph so that each neighboring vertex has a different color. The minimum number of colors used to color the vertices of a graph is called the chromatic number. In this article, we discuss the chromatic number in the amalgamated graph of wheel graph and star graph and the chromatic number on the amalgamation of wheel graph and cycle graph. The results obtained are as follows: (1) the chromatic number of the amalgamated graph of the wheel graph () and the star graph () for n even is 3 and for n odd is 4, (2) the chromatic number of the graph resulting from the operation amalgamation of the wheel graph () and the cycle graph () for n even is 3 and for odd is 4.</p>

Accelerating Backward Aggregation in GCN Training With Execution Path Preparing on GPUs

The emerging Graph Convolutional Network (GCN) has now been widely used in many domains, and it is challenging to improve the efficiencies of applications by accelerating the GCN trainings. For the sparsity nature and exploding scales of input real-world graphs, state-of-the-art GCN training systems (e.g., GNNAdvisor) employ graph processing techniques to accelerate the message exchanging (i.e. aggregations) among the graph vertices. Nevertheless, these systems treat both the aggregation stages of forward and backward propagation phases as all-active graph processing procedures that indiscriminately conduct computation on all vertices of an input graph. In this paper, we first point out that in a GCN training problem with a given training set, the aggregation stages of its backward propagation phase (called as backward aggregations in this paper) can be converted to partially-active graph processing procedures, which conduct computation on only partial vertices of the input graph. By leveraging such a finding, we propose an execution path preparing method that collects and coalesces the data used during backward propagations of GCN training conducted on GPUs. The experimental results show that compared with GNNAdvisor, our approach improves the performance of the backward aggregation of GCN trainings on typical real-world graphs by 1.48x~5.65x. Moreover, the execution path preparing can be conducted either before the training (during preprocessing) or on-the-fly with the training. When used during preprocessing, our approach improves the overall GCN training by 1.05x~1.37x. And when used on-the-fly, our approach improves the overall GCN training by 1.03x~1.35x.

The extreme vertices of the power graph of a group

For a fixed finite group G, the power graph of G was defined to be the simple graph Γ(G) whose vertex set V(Γ(G))=G, and edge set E(Γ(G))={xy: either x=yn or y=xn for some integer n}. In this paper the extreme vertices of the power graph of abelian groups, dihedral groups and dicyclic groups have been characterized.

Large Rainbow Cliques in Randomly Perturbed Dense Graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every proper coloring of its edges yields a rainbow copy of $H$. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every $s$. In this paper, we show that for $s \geq 9$ the threshold is $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 \leq s \leq 7$, the threshold is lower; see our companion paper for more details. Also in this paper, we determine that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$ is $n^{-2}$ for every $\ell \geq 2$; in particular, the threshold does not depend on the length of the cycle $C_{2\ell - 1}$. For even cycles, and in fact any fixed bipartite graph, no random edges are needed at all; that is, $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ always holds, whenever $G$ is as above and $H$ is bipartite.

The Distance Orientation Problem

The Distance Orientation Problem (DOP) is formulated as follows: Given a graph with positive weights on its edges, are there weights for the vertices, such that for every edge x y it holds that the absolute difference between the weights of x and y is equal to the weight of x y ? This problem can also be formulated as a problem of finding a special orientation of G and was motivated by an application in Shape from Shading, a method in the field of Computer Vision. We present a linear-time algorithm for complete 3-cover graphs, a generalization of chordal graphs and planar triangulations. For outerplanar graphs we show that the DOP is fixed-parameter tractable and present a pseudo-polynomial time algorithm for integral edge weights. Both algorithms use the idea that the existence of feasible weights for the vertices of an outerplanar graph can be decided by only looking at the edge weights of each face of its outerplane embedding separately. We show that this property does not hold for any embedding of a planar graph which is not outerplanar. Furthermore, we prove that the DOP is strongly NP -complete for grid graphs, i.e., there is no pseudo-polynomial algorithm to solve it unless P = NP .

Cell-Based Synthesis of Multiple Analog Filter and Oscillator Topologies Employing Graph

A new synthesis methodology, which offers design harmony of multiple analog filter and oscillator topologies is presented. Vertices of a bipartite graph are defined with simple mathematical function, denoted as cell. Weight of the edges are denoted by relation operator. A general theory has been developed based on graph and a set of traversal rules conjugated with cell, to produce multiple number of second-order function (SOF) in terms of cells. Theoretical approach has been facilitated with operational transconductance amplifier (OTA)-based circuit realization of multiple filter and oscillator topologies, and abridged with algorithm, which may lead to virtual EDA tool and field programmable analog array (FPAA). The proposed synthesis approach is validated with SPICE simulation.

A note on eigenvalue, spectral radius and energy of extended adjacency matrix

The extended adjacency matrix of a graph has been introduced to expand some beneficial molecular topological indices. In the present work, the spectra of certain classes of vertex or edge-transitive graphs are investigated. Also, we study some spectral properties of extended adjacency matrix. Besides, we study the spectral A e x -spread of a graph. In this way, we explore some new bounds for both the smallest and the largest eigenvalues. Finally, we persue the behaviour of A e x -energy of a graph.

Пошук максимальних незалежних множин вершин графа для вдосконалення програмних проєктів

The need is fixed to software project enhancing with seamless integration of technological-descriptive and normative project manage- ment approaches by means of classical Graph Discrete Optimization Problems tailoring for software project management tasks, poorly equipped with best practices within technological approach. Class of software project management tasks is proposed to demonstrate the benefits of such integration. Two Boolean linear programming problems are investigated for searching some maximum size indepen- dent set (Section 1) and an algorithm for searching all possible maximum size independent sets (Section 2). Section 3 presents Problem Statement for searching a given number of non-intersecting independent sets with maximum sum of vertices’ numbers within independent sets. Based on it, Vizing-Plesnevich algorithm is described for coloring the graph vertices with the minimum number of colors. To solve Boolean problems, both specialized mathematical programming language AMPL and corresponding solver program named gu- robi are used. For basic algorithms developed, reference AMPL code versions are given as well as their running results. Illustrative examples of software project enhancing with the algorithms elaborated are considered in Section 4, namely: 25 specialists being conflicted during their previous projects partitioning into coherent conflict-free sub-teams for software projects portfolio; schedule optimization for autonomous testing of reusable components within a critical software system; cores composing for independent teams in a critical software project.

Upper bounds on the smallest positive eigenvalue of trees with at most one zero eigenvalue

In this article, we consider the trees for which the adjacency matrix has nullity at most one. We study the smallest positive (adjacency) eigenvalue of these trees. For such trees lower bounds on the smallest positive eigenvalue have already been obtained in literature but the problem of finding an upper bound still remains unsolved. In this article, we find upper bounds on the smallest positive eigenvalue of these trees and obtain corresponding unique maximal trees. Additionally, we study the smallest positive eigenvalue of trees obtained by attaching a pendant at some vertex of a path graph and also identify the trees for the extremal cases. As applications of our earlier results, we show that, for a path of odd order, the smallest positive eigenvalue is maximum, if we add the pendant almost in the middle. As another application, we show that for a path of any order, the smallest positive eigenvalue is minimum, if we add the pendant vertex at the end.

Arithmetic Sequential Graceful Labeling on Star Related Graphs

Objectives: To identify a new family of Arithmetic sequential graceful graphs. Methods: The methodology involves mathematical formulation for labeling of the vertices of a given graph and subsequently establishing that these formulations give rise to arithmetic sequential graceful labeling. Findings: In this study, we analyzed some star related graphs namely Star graph, Ustar, t-star, and double star proved that these graphs possess Arithmetic sequential graceful labeling. Novelty: Here, we introduced a new labeling called Arithmetic sequential graceful labeling and we give Arithmetic sequential graceful labeling to some star related graphs. Keywords: Star graph; t-star; U-star; Graceful labeling; Double star

Arbitrary pattern formation on infinite regular tessellation graphs

Given a set R of robots, each one located at a different vertex of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that |R|=|F|, APF asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections.So far, as possible graph discretizing the Euclidean plane only the standard square grid has been considered in the context of the classical OBLOT model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. In particular, the former can be considered as the most general in terms of possible symmetries and trajectories.We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph. The algorithm and its correctness are based on a rigorous methodology.

Order based algorithms for the core maintenance problem on edge-weighted graphs

The cohesive subgraph k-core is a maximal connected subgraph with the minimum degree δ≥k of a simple graph, where integer k≥0. Define the core number of a vertex w as the maximum k such that w is contained in a k-core. The core decomposition problem which is calculating the core numbers of all vertices in static graphs, and the core maintenance problem which is updating the core numbers in dynamic graphs are our main concern. Although, core numbers can be updated by the core decomposition algorithms, only a small part of vertices' core numbers have changed after the change of a graph. Thus, it is necessary to update core numbers locally to reduce the cost. In this paper, we study the core maintenance problem on edge-weighted graphs by using the vertex sequence k-order which is ordered by the order that the core decomposition algorithm removes vertices. We design the core maintenance algorithms for inserting one edge at a time and the method of updating the k-order, which reduce the searching range and the time cost evidently. For the removing case, we use the existing subcore algorithm to do the core maintenance and modify it with the method of updating k-order we design. Finally, we do extensive experiments to evaluate the effectiveness and the efficiency of our algorithms, which shows that the order based algorithm has a better performance than the existing for the core maintenance on edge-weighted graphs.

The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity e2πik (and its symmetric entry is e−2πik); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set {1,e2πik,e−2πik}, which is not closed under multiplication when k⩾4. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank (Hk-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with Hk-rank 2 and 3, respectively.

Equitable critical graphs

A proper vertex coloring of a graph [Formula: see text] is equitable if the sizes of any two color classes differ by at most one. The equitable chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] is equitably [Formula: see text]-colorable. In this paper, we discuss some basic properties of equitable critical graphs as well as equitable [Formula: see text]-critical graphs. A graph [Formula: see text] is called equitable critical if [Formula: see text] for every proper subgraph [Formula: see text] of [Formula: see text]. [Formula: see text] is called equitable [Formula: see text]-critical if it is equitable [Formula: see text]-chromatic and equitable critical. Furthermore, we discuss that equitable vertex (edge) critical, equitable critical vertex (edge) graphs.