Vertices Of Graph Research Articles

RECONSTRUCTION THE DOMAIN OF DEFINITION OF SOMEDIFFERENTIAL OPERATOR ON A DIRECTED GRAPH

In this work it is proposed to study differential operators on a graph as an operator composed of differential operators on one-dimensional arcs and matrix operators on interior vertices of the graph. The work explores some questions concerning the theoretical side of ordinary differential equations with integro-differential conditions on stratified sets like graph. The attention will be paid to reconstruction of the domain of differential operator on directed graph. The reconstruction of the domain of differential operator means a simple specifying the boundary conditions from a known differential equations and its known eigenvalues. The paper studies the case of the second order differential equations with irregular boundary conditions on the vertices of directed graph. To achieve our goal we use the fact that finite set of eigenvalues serves as additional information for reconstruction of the domain of the differential operator on stratified set. The constructive algorithms for reconstructing the domain of definition of differential operator on directed graph are developed. All boundary functions from the spectral data are uniquely restored.

On RAC Drawings of Graphs with Two Bends per Edge

It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.

A novel edge-weighted matrix of a graph and its spectral properties with potential applications

<p>Regarding a simple graph $ \Gamma $ possessing $ \nu $ vertices ($ \nu $-vertex graph) and $ m $ edges, the vertex-weight and weight of an edge $ e = uv $ are defined as $ w(v_{i}) = d_{ \Gamma}(v_{i}) $ and $ w(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2 $, where $ d_{ \Gamma}(v) $ is the degree of $ v $. This paper puts forward a novel graphical matrix named the edge-weighted adjacency matrix (adjacency of the vertices) $ A_{w}(\Gamma) $ of a graph $ \Gamma $ and is defined in such a way that, for any $ v_{i} $ that is adjacent to $ v_{j} $, its $ (i, j) $-entry equals $ w(e) = d_{ \Gamma}(v_{i})+d_{ \Gamma}(v_{j})-2 $; otherwise, it equals 0. The eigenvalues $ \lambda_{1}^{w}\ge \lambda_{2}^{w}\ge\ldots\ge \lambda_{\nu}^{w} $ of $ A_w $ are called the edge-weighted eigenvalues of $ \Gamma $. We investigate the mathematical properties of $ A_{w}(\Gamma) $'s spectral radius $ \lambda_{1}^{w} $ and energy $ E_{w}(\Gamma) = \sum_{i = 1}^{\nu}|\lambda_{i}^{w}| $. Sharp lower and upper bounds are obtained for $ \lambda_{1}^{w} $ and $ E_{w}(\Gamma) $, and the respective extremal graphs are characterized. Further, we employ these spectral descriptors in structure-property modeling of the physicochemical properties of polycyclic aromatic hydrocarbons for a set of benzenoid hydrocarbons (BHs). Detailed regression analysis showcases that edge-weighted energy outperforms classical adjacency energy in structure-property modeling of the physicochemical properties of BHs.</p>

ResGEM: Multi-scale Graph Embedding Network for Residual Mesh Denoising.

Mesh denoising is a crucial technology that aims to recover a high-fidelity 3D mesh from a noise-corrupted one. Deep learning methods, particularly graph convolutional networks (GCNs) based mesh denoisers, have demonstrated their effectiveness in removing various complex real-world noises while preserving authentic geometry. However, it is still a quite challenging work to faithfully regress uncontaminated normals and vertices on meshes with irregular topology. In this paper, we propose a novel pipeline that incorporates two parallel normal-aware and vertex-aware branches to achieve a balance between smoothness and geometric details while maintaining the flexibility of surface topology. We introduce ResGEM, a new GCN, with multi-scale embedding modules and residual decoding structures to facilitate normal regression and vertex modification for mesh denoising. To effectively extract multi-scale surface features while avoiding the loss of topological information caused by graph pooling or coarsening operations, we encode the noisy normal and vertex graphs using four edge-conditioned embedding modules (EEMs) at different scales. This allows us to obtain favorable feature representations with multiple receptive field sizes. Formulating the denoising problem into a residual learning problem, the decoder incorporates residual blocks to accurately predict true normals and vertex offsets from the embedded feature space. Moreover, we propose novel regularization terms in the loss function that enhance the smoothing and generalization ability of our network by imposing constraints on normal consistency. Comprehensive experiments have been conducted to demonstrate the superiority of our method over the state-of-the-art on both synthetic and real-scanned datasets.

Some approximations and identities from special sequences for the vertices of suborbital graphs

Some approximations and identities from special sequences for the vertices of suborbital graphs

Monotonic Entropies

We introduce an axiomatization of entropy that generates, as special cases, novel entropy types. These entropies generalize Shannon’s entropy and allow the introduction of entropy for partitions of sets of objects located in metric spaces, and for partitions of sets of vertices in undirected graphs. Corresponding metrics on the sets of partitions are introduced. Also, we hint to applications of these metrics for evaluation of clustering quality.

Nash Equilibria in Reverse Temporal Voronoi Games

We study Voronoi games on temporal graphs as introduced by Boehmer et al. (IJCAI '21) where two players each select a vertex in a temporal graph with the goal of reaching the other vertices earlier than the other player. In this work, we consider the reverse temporal Voronoi game, that is, a player wants to maximize the number of vertices reaching her earlier than the other player. Since temporal distances in temporal graphs are not symmetric in general, this yields a different game. We investigate the difference between the two games with respect to the existence of Nash equilibria in various temporal graph classes including temporal trees, cycles, grids, cliques and split graphs. Our extensive results show that the two games indeed behave quite differently depending on the considered temporal graph class.

Normalized ground states for a doubly nonlinear Schrödinger equation on periodic metric graphs

We investigate the existence of ground states for a class of Schrödinger equations with both a standard power nonlinearity and delta nonlinearity concentrated at finite vertices of the periodic metric graphs $ G $. Using variational methods, if $ \alpha > 0 $ and the standard nonlinearity power is $ L^{2}- $subcritical, we establish the existence of ground states for every mass and every periodic graph. If $ \alpha < 0 $ and the standard nonlinearity power is $ L^{2}- $critical, we show that two types of topological structures on $ G $ will prevent the existence of ground states. Furthermore, for graphs that do not satisfy these two types of topological structures, ground states exist when the given mass belongs to an appropriate range and the parameter $ \left | \alpha \right| $ is small enough.

Cloud-based mathematical models for self-organizing swarms of UAVs: design and analysis

Unmanned aerial vehicle (UAV) swarms have gained significant attention for their potential applications in various fields. The effective coordination and control of UAV swarms require the development of robust mathematical models that can capture their complex dynamics. The paper introduces mathematical models and relevant paradigms based on the design and analysis of self-organizing swarms of UAVs. The logical and technological construction of the model relies on the theorems developed by authors for obtaining full information exchange during the swarm quasi-random walk. The suggested rotor-router model interprets the discrete-time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on the vertices of the swarm graph. The recommended optimal and fault-tolerant gossip/broadcast schemes support the resilience of swarm to internal failures and external attacks, and cryptographic protocols approve the security. The proposed cloud network topology serves as the implementation framework for the model, encompassing various connectivity options to ensure the expected behavior of the UAV swarms.

Route planning of autonomous robots in three-dimensional logic space using mivar technologies

The paper describes an approach to using mivar technologies for planning three-dimensional robot routes, taking into account the given obstacles. This system is of great importance in increasing the autonomy of robots, as it will ultimately help us get closer to creating artificial intelligence in mechanical engineering. At present, there are many tasks related to automation and robotics that require non-trivial solutions. The article describes three models of three-dimensional logical space and presents a visualization of route construction in these spaces for robotic complexes. Obstacles in three-dimensional logical space are understood as the absence of graph vertices and transitions between them in a certain specific area. The use of mivar technologies of logical artificial intelligence in the field of transport systems allows to significantly speed up route planning thanks to a unique mivar algorithm for processing information, which allows creating systems capable of making decisions in real time. This work is intended for researchers dealing with the problem of three-dimensional route planning using mivar technologies.

Certain Types of Vertices in m-Polar Fuzzy Graphs

In this study, we explore the concept of m-polar fuzzy (mPF) detour g-eccentric nodes within m-polar fuzzy graphs (mPFGs). We delve into the idea of mPF detour g-interior nodes and mPF detour g-boundary nodes, examining their significance and properties. Additionally, we establish the relationship between mPF detour g-boundary vertices and mPF cut vertices.

A method of secure network traffic routing based on specified criterias

Due to the implementation of new network services, the increase amount of data that need to be transmitted, and the use of networks in various sectors with diverse communication requirements, there is a need to develop new approaches to ensure the quality of such communications. Leading network equipment manufacturers and standardization organizations are developing new routing algorithms, resulting in the introduction of new routing protocols or improvements to existing ones. However, all these algorithms cover routing principles for general-purpose networks and do not consider the communication requirements of specialized networks. Therefore, the task arises to research optimization directions for network traffic routing, define optimization criteria, and further develop a method for secure network traffic routing based on the specified criteria. In this work, a routing method is proposed that takes into account the defined requirements when searching for the optimal route. In the case of dynamic routing, each router calculates the shortest routes to all other networks based on the shortest path search algorithm. This work defines a method for calculating metrics based on specified criteria and formally describes the algorithm for finding the shortest path. Quality of communication criteria is introduced, which will enable meeting communication requirements in specialized networks. Calculation methods for these criteria are demonstrated, and data collection methods for the calculation of specified criteria are determined. A formula for calculating metrics is proposed, which includes the possibility of selecting T-values and determining their numerical parameters to prioritize specific criteria. Default values for criteria are defined, and metric calculations are tested by default for different types of interfaces. After calculating metrics, the task reduces to finding the shortest paths in a weighted graph using an algorithm based on Dijkstra's algorithm. The proposed algorithm for finding the shortest path involves identifying the primary (shortest) and backup paths from a given source vertex to all other graph vertices. A formal description of the proposed algorithm is provided.

New Spectral Results for Laplacian Harary Matrix and the Harary Laplacian-Energy-like Applying a Matrix Order Reduction

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs.

Greedy maximal independent sets via local limits

AbstractThe random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences of (possibly random) graphs by employing a notion of local convergence. We use this framework to give straightforward proofs for results on previously studied families of graphs, like paths and binomial random graphs, and to study new ones, like random trees and sparse random planar graphs. We conclude by analysing the random greedy algorithm more closely when the base graph is a tree.

Flip : Data-centric Edge CGRA Accelerator

Coarse-Grained Reconfigurable Arrays (CGRA) are promising edge accelerators due to the outstanding balance in flexibility, performance, and energy efficiency. Classic CGRAs statically map compute operations onto the processing elements (PE) and route the data dependencies among the operations through the Network-on-Chip. However, CGRAs are designed for fine-grained static instruction-level parallelism and struggle to accelerate applications with dynamic and irregular data-level parallelism, such as graph processing. To address this limitation, we present Flip , a novel accelerator that enhances traditional CGRA architectures to boost the performance of graph applications. Flip retains the classic CGRA execution model while introducing a special data-centric mode for efficient graph processing. Specifically, it leverages the inherent data parallelism of graph algorithms by mapping graph vertices onto PEs rather than the operations and supporting dynamic routing of temporary data according to the runtime evolution of the graph frontier. Experimental results demonstrate that Flip achieves up to 36× speedup with merely 19% more area compared to classic CGRAs. Compared to state-of-the-art large-scale graph processors, Flip has similar energy efficiency and 2.2× better area efficiency at a much-reduced power/area budget.

Clique factors in pseudorandom graphs

An n -vertex graph is said to to be (p,\beta) -bijumbled if for any vertex sets A,B\subseteq V(G) , we have e(A,B)=p|A|\,|B|\pm \beta \sqrt{|A|\,|B|}. We prove that for any r\in {\mathbb N}_{\ge3} and c>0 there exists an \varepsilon>0 such that any n -vertex (p,\beta) -bijumbled graph with n\in r \mathbb{N} , p>0 , \delta(G)\geq cpn and \beta \leq \varepsilon p^{r-1}n contains a K_{r} -factor. This implies a corresponding result for the stronger pseudorandom notion of (n,d,\lambda) -graphs. For the case of triangle factors, that is, when r=3 , this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of \beta=o(p^{2}n) actually guarantees that a (p,\beta) -bijumbled graph G contains every graph on n vertices with maximum degree at most 2.

Game problem of assigning staff to project implementation

This article describes how to solve the game problem of assigning staff to work on projects based on the ontological approach. The stochastic game algorithm for colouring an undirected random graph has been used to plan project implementation. The stochastic game mathematical model has been described, and the self-learning Markov method has been used for its solution. It is highlighted that the goal of the players is to minimize the functions of average losses. The Markov recurrent method that provides the adaptive choice of colours for the vertices of the random graph based on dynamic vectors of mixed strategies, the values of which depend on the current losses of players has been used. A computer experiment was carried out, which confirmed the convergence of the stochastic game for the problem of colouring the random graph. In conclusion. the possibility of defining the procedure for appointing staff to implement projects has been justified.

Multiscale transforms for signals on simplicial complexes

Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.

5‐Coloring reconfiguration of planar graphs with no short odd cycles

AbstractThe coloring reconfiguration graph has as its vertex set all the proper ‐colorings of , and two vertices in are adjacent if their corresponding ‐colorings differ on a single vertex. Cereceda conjectured that if an ‐vertex graph is ‐degenerate and , then the diameter of is . Bousquet and Heinrich proved that if is planar and bipartite, then the diameter of is . (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when is planar and has no 3‐cycles. As a partial solution to this problem, we show that the diameter of is for every planar graph with no 3‐cycles and no 5‐cycles.

Теорема існування для задачі масопереносу на графі

The mass transfer process in a porous medium is described using the Richards-Klute equation. This equation describes mass flows due to the actions of gravity and capillarity and allows modeling the mass transfer process with saturation limit. The Richards-Klute equation is a nonlinear elliptic-parabolic partial differential equation, so the main methods for solving it and modeling the mass transfer process are numerical methods. The article considers a model of a system of interconnected pipes, inside which the process of mass transfer takes place. Such systems are often found in agriculture and are actively used in the construction of irrigation systems. The article proposes to model pipe system using graphs, where pipes are represented by graph edges, and connection points or free ends of system pipes are represented by graph vertices. The article contains the definitions of the Richards-Klute equation on a graph in the usual and weak forms. On the edges of the graph, one-dimensional cases of the Richards-Klute equation are considered, while on the vertices either the boundary conditions are given or the equation that models the law of mass conservation is given. The definitions of the solution and weak solution of the Richards-Klute equation on the graph are also given. Also, the theorem of the existence of a weak solution of the Richards-Klute equation on a graph is proved. To prove the theorem of the existence of a weak solution of the Richards-Klute equation on a graph, the Kirchhoff transformation is used and conditions are given that are analogous to the conditions used in the proof of the existence of a weak solution of the Richards-Klute equation in regular domains in three-dimensional space, and which are defined in the classical work [1], which is devoted to the problems of existence and uniqness of weak solutions of elliptic-parabolic partial differential equations.