Graph Distance Research Articles

Formula omitted]-packing colorings of distance graphs [formula omitted

Given a graph G and a non-decreasing sequence S=(a1,a2,…) of positive integers, the mapping f:V(G)→{1,…,k} is an S-packing k-coloring of G if for any distinct vertices u,v∈V(G) with f(u)=f(v)=i the distance between u and v in G is greater than ai. The smallest k such that G has an S-packing k-coloring is the S-packing chromatic number, χS(G), of G. In this paper, we continue the study of S-packing colorings of distance graphs, initiated by Togni (2014) and Ekstein et al. (2012). We focus on the distance graphs G(Z,{2,t}), where t>1 is an odd integer, which has Z as its vertex set, and i,j∈Z are adjacent if |i−j|∈{2,t}. We determine the S-packing chromatic numbers of the graphs G(Z,{2,t}), where S is any sequence with ai∈{1,2} for all i. In addition, we give lower and upper bounds for the d-distance chromatic numbers of the distance graphs G(Z,{2,t}), which in the cases d≥t−3 give the exact values. Implications for the corresponding S-packing chromatic numbers of the circulant graphs are also discussed.

Study of Circular Distance in Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖 P〗_2×C_n. Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed.

Design and manufacturing of Shock pot mounts for Road Quality Analysis

The main purpose of this project was to design and fabricate a system that can be used for road quality analysis. It incorporates a linear potentiometer sensor to sense the displacement of the wheel, a data logger module to collect and store the sensor data, and a mechanical mount to hold the sensor rigidly in the desired position. The sensor data stored in the datalogger is then filtered and smoothened using Butterworth filter to get clean data. This data is further plotted on a displacement v/s distance graph. Standard color codes are decided and are assigned to each condition based on their ratios. Color codes, therefore, helps in the quick graphical interpretation of the data. The solid model of the mounts is done on solid modelling software and the data interpretation part is done using MATLAB. The proposed system can be mounted on various vehicles and can be further improved by analysing the live data and representing them on the navigation systems.

Bounds on Borsuk Numbers in Distance Graphs of a Special Type

Bounds on Borsuk Numbers in Distance Graphs of a Special Type

Random walk on random planar maps: Spectral dimension, resistance and displacement

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a “mating-of-trees” type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the γ-mated-CRT map for γ∈(0,2). For each of these maps, we obtain an upper bound for the Green’s function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after n steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors. When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after 2n steps is n−1+on(1). Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least n1/4−on(1) units of graph distance in n units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (Probab. Theory Related Fields 178 (2020) 567–611). These two works together resolve a conjecture of Benjamini and Curien (Geom. Funct. Anal. 23 (2013) 501–531) in the UIPT case. Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.

A Quasi-Hole Detection Algorithm for Recognizing k-Distance-Hereditary Graphs, with k < 2

Cicerone and Di Stefano defined and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the k-distance-hereditary graphs such that k<2. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.

A Polynomial Kernel for Distance-Hereditary Vertex Deletion

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph G on n vertices and an integer k, whether there is a set S of at most k vertices in G such that $$G-S$$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width because distance-hereditary graphs are exactly the graphs of rank-width at most 1. Eiben, Ganian, and Kwon (JCSS’ 18) proved that Distance-Hereditary Vertex Deletion can be solved in time $$2^{{\mathcal {O}}(k)}n^{{\mathcal {O}}(1)}$$ , and asked whether it admits a polynomial kernelization. We show that this problem admits a polynomial kernel, answering this question positively. For this, we use a similar idea for obtaining an approximate solution for Chordal Vertex Deletion due to Jansen and Pilipczuk (SIDMA’ 18) to obtain an approximate solution with $${\mathcal {O}}(k^3\log n+ k^2\log ^2 n)$$ vertices when the problem is a Yes-instance, and we exploit the structure of split decompositions of distance-hereditary graphs to reduce the total size.

Fractional diffusion maps

In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with β -stable Lévy processes. For local kernels, while the method is a version of the diffusion maps algorithm, we show that the applications with non-Gaussian local heat kernels approximate temporally rescaled Laplace-Beltrami operators. For the non-local heat kernels, we modify the diffusion maps algorithm to estimate fractional Laplacian operators. Here, the graph distance is used to approximate the geodesic distance with appropriate error bounds. While this approximation becomes numerically expensive as the number of data points increases, it produces an accurate operator estimation that is robust to the choice of the kernel bandwidth parameter value. In contrast, the local kernels are numerically more efficient but more sensitive to the choice of kernel bandwidth parameter value. In an application to estimate non-smooth regression functions, we find that using the nonlocal kernel as a regularizer produces a more robust and accurate estimate than using local kernels. For manifolds with boundary, we find that the proposed fractional diffusion maps framework implemented with non-local kernels approximates the regional fractional Laplacian.

Dual distance adaptive multiview clustering

Adaptively adjusting the graph, by taking the clustering capability into consideration, has become popular in graph-based clustering methods and extended to multiview clustering problem. Existing methods learn the graph from pairwise distances of the data in the original space or a linearly projected space, which requires that those representations can finely reflect the implicit data structure. However, the data structure in high-dimensional space may not always lie on a linear manifold, and this problem becomes more critical in multiview conditions. Aim at this, we propose a multiview clustering method based on adaptive graph and dual distance. Specifically, we fuse the distances computed from nonlinear embedding space and original space. An adaptive graph is then constructed based on the fused distance, and it is more reliable for multiview clustering. In our approach, the clustering result with exact number of clusters can be found without post-processing. We evaluate the proposed method on several multiview datasets, the experimental results show our approach is superior to the state-of-the-art multiview clustering methods.

Applying Class Distance to Decide Similarity on Information Models for Automated Data Interoperability

In the world of the Internet of Things (IoT), heterogeneous systems and devices need to be connected and exchange data with others. How data exchange can be automatically realized becomes a critical issue. An information model (IM) is frequently adopted and utilized to solve the data interoperability problem. Meanwhile, as IoT systems and devices can have different IMs with different modeling methodologies and formats such as UML, IEC 61360, etc., automated data interoperability based on various IMs is recognized as an urgent problem. In this paper, we propose an approach to automate the data interoperability, i.e. data exchange among similar entities in different IMs. First, similarity scores among entities are calculated based on their syntactic and semantic features. Then, in order to precisely get similar candidates to exchange data, a concept of class distance calculated with a Virtual Distance Graph (VDG) is proposed to narrow down obtained similar properties for data exchange. Through analyzing the results of a case study, the class distance based on VDG can effectively improve the precisions of calculated similar properties. Furthermore, data exchange rules can be generated automatically. The results reveal that the approach of this research can efficiently contribute to resolving the data interoperability problem.

Renyi entropy driven hierarchical graph clustering.

This article explores a graph clustering method that is derived from an information theoretic method that clusters points in n}{}{{mathbb{R}}^{n}} relying on Renyi entropy, which involves computing the usual Euclidean distance between these points. Two view points are adopted: (1) the graph to be clustered is first embedded into n}{}{mathbb{R}}^{d} for some dimension d so as to minimize the distortion of the embedding, then the resulting points are clustered, and (2) the graph is clustered directly, using as distance the shortest path distance for undirected graphs, and a variation of the Jaccard distance for directed graphs. In both cases, a hierarchical approach is adopted, where both the initial clustering and the agglomeration steps are computed using Renyi entropy derived evaluation functions. Numerical examples are provided to support the study, showing the consistency of both approaches (evaluated in terms of F-scores).

Metric dimension of critical Galton–Watson trees and linear preferential attachment trees

The metric dimension of a graph G is the minimal size of a subset R of vertices of G that, upon reporting their graph distance from a distinguished (source) vertex v⋆, enable unique identification of the source vertex v⋆ among all possible vertices of G. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton–Watson trees conditioned to have size n, and growing general linear preferential attachment trees. The former class includes uniform random trees, the latter class includes Yule-trees (also called random recursive trees), m-ary increasing trees, binary search trees, and positive linear preferential attachment trees. In all these cases, we are able to identify the limiting constant in the LLN explicitly. Our result relies on the insight that the metric dimension can be related to subtree properties, and hence we can make use of the powerful fringe-tree literature developed by Aldous and Janson et al.

Distances in graphs of girth 6 and generalised cages

In this paper we present bounds on the radius and diameter of graphs of girth at least 6 and for (C4,C5)-free graphs, i.e., graphs not containing cycles of length 4 or 5. We show that the diameter of a graph G of girth at least 6 is at most 3nδ2−δ+1−1, and the radius is at most 3n2(δ2−δ+1)+10, where n is the order and δ the minimum degree of G. If δ−1 is a prime power, then both bounds are sharp apart from an additive constant.For graphs of large maximum degree Δ, we show that these bounds can be improved to 3n−Δδδ2−δ+1−3(δ−1)Δ(δ−2)δ2−δ+1+10 for the diameter, and 3n−3Δδ2(δ2−δ+1)−3(δ−1)Δ(δ−2)2(δ2−δ+1)+22 for the radius. We further show that only slightly weaker bounds hold for (C4,C5)-free graphs.As a by-product we obtain a result on a generalisation of cages. For given δ,Δ∈N with Δ≥δ let n(δ,Δ,g) be the minimum order of a graph of girth g, minimum degree δ and maximum degree Δ. Then n(δ,Δ,6)≥Δδ+(δ−1)Δ(δ−2)+32. If δ−1 is a prime power, then there exist infinitely many values of Δ such that, for δ constant and Δ large, n(δ,Δ,6)=δΔ+O(Δ).

DRGraph: An Efficient Graph Layout Algorithm for Large-scale Graphs by Dimensionality Reduction.

Efficient layout of large-scale graphs remains a challenging problem: the force-directed and dimensionality reduction-based methods suffer from high overhead for graph distance and gradient computation. In this paper, we present a new graph layout algorithm, called DRGraph, that enhances the nonlinear dimensionality reduction process with three schemes: approximating graph distances by means of a sparse distance matrix, estimating the gradient by using the negative sampling technique, and accelerating the optimization process through a multi-level layout scheme. DRGraph achieves a linear complexity for the computation and memory consumption, and scales up to large-scale graphs with millions of nodes. Experimental results and comparisons with state-of-the-art graph layout methods demonstrate that DRGraph can generate visually comparable layouts with a faster running time and a lower memory requirement.

Proof of a conjecture on spectral distance between cycles, paths and certain trees

We confirm the following conjecture which has been proposed by Stanić in [I. Jovanović and Z. Stanić, Spectral distances of graphs, Linear Algebra Appl. 436(5) (2012) 1425–1435]: [Formula: see text] [Formula: see text] where [Formula: see text] is the spectral distance between [Formula: see text] vertex non-isomorphic graphs [Formula: see text] and [Formula: see text] with adjacency spectra [Formula: see text] for [Formula: see text], and [Formula: see text] and [Formula: see text] denote the path and cycle on [Formula: see text] vertices, respectively, [Formula: see text] denotes the coalescence of [Formula: see text] and [Formula: see text] on one of the vertices of degree 1 of [Formula: see text] and the vertex of degree [Formula: see text] of [Formula: see text], and [Formula: see text] denotes the coalescence of [Formula: see text] and [Formula: see text] on the vertex of degree 1 of [Formula: see text] which is adjacent to a vertex of degree [Formula: see text] and the vertex of degree [Formula: see text] of [Formula: see text].

Resistance Distances in Linear Polyacene Graphs

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. In this article, using electric network approach and combinatorial approach, we derive exact expression for resistance distances between any two vertices of polyacene graphs.

Distance measures for embedded graphs

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded and if the embedding meets a certain geometric restriction. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.

Minimum Degree Distance of Five Cyclic Graphs

Let G be a connected graph with n vertices. Then the class of connected graphs having n vertices is denoted by Gn. The subclass of connected graphs with 5 cycles are denoted by Gn5. The classification of graph G∈Gn5 depends on the number of edges and the sum of the degrees of the vertices of the graph. Any graph in Gn5 contains five linearly independent cycles having at least n+3 edges and the sum of degrees of vertices of 5-cyclic must be equal to twice of n+4. In this paper, minimum degree distance of class of five cyclic connected graph is investigated. To find minimum degree distance of a graph some transformations T have been defined. These transformation have been applied on the graph G∈Gn5 in such a way that the resultant graph belongs to Gn5 and also degree distance of T(G) is always must be less than G. For n=5, the five 5-cyclic graph has minimum degree distance 78 and the minimum degree distance of 5-cyclic graphs having six vertices is 124. In case of n greater than 6, a general formula for minimum degree distance is investigated. In this paper, we proved that the minimum degree distance of connected 5 cyclic graphs is 3n2+13n-62 by using transformations, for n≥7.

On the estimation of latent distances using graph distances

We are given the adjacency matrix of a geometric graph and the task of recovering the latent positions. We study one of the most popular approaches which consists in using the graph distances and derive error bounds under various assumptions on the link function. In the simplest case where the link function is proportional to an indicator function, the bound matches an information lower bound that we derive.

МЕТОДИКА СИНТЕЗА ПОДСИСТЕМЫ ИНТЕЛЛЕКТУАЛЬНОГО МОНИТОРИНГА ИНФОРМАЦИОННО}ТЕЛЕКОММУНИКАЦИОННОЙ СЕТИ СИТУАЦИОННОГО ЦЕНТРА

Introduction: in accordance with the task of forming a system of distributed situational centers of public authorities, it is necessary to interface heterogeneous segments of the department's information and telecommunications networks into a geographically distributed infrastructure with the construction of a subsystem for monitoring the state of its elements on it. The purpose of the study: based on the use of basic concepts of structural synthesis methods, structural analysis of network infrastructures, as well as parametric synthesis procedures, to develop a methodology for the synthesis of a new generation monitoring subsystem based on intelligent methods for identifying the state of the network. Results: the structuring of the controlled space by the terms "monitoring zone"," critical element"," state classes "form the basis of new methods of intelligent monitoring that are" insensitive " to the properties of the constant evolution of network infrastructures. The proposed method of synthesis of the intelligent monitoring subsystem consists of sequentially performed stages of structural synthesis, parametric synthesis and structural analysis of the network. Practical significance: the successive stages of the methodology using graph distance measurement procedures and the modified k-means algorithm allow not only to identify the type of network state, but also to reasonably, using instrumental calculation methods, present in the interests of the decision support system sets of acceptable values of the main parameters and probabilistic-temporal characteristics of the monitoring subsystem for subsequent reconfiguration of the network and preventing its transition to an inoperable state. Discussion: the modification of the k-means algorithm proposed in the study differs in that in the classical algorithm, work is carried out on points of Euclidean space, and in the proposed method we are talking about a graph space with metrics in the form of graph distances, while data clouds are used as initial data for classification in the k-means algorithm as disordered data sets that are not tied to any of the measurement scales, and in the proposed algorithm, the data cloud is represented by a set of graphs in a given topological space of graph metrics, describing the state of the network in time.