Graph Distance Research Articles

On sum of powers of normalized Laplacian eigenvalues and resistance distances of graphs

Let G=(V,E) be a simple connected graph with vertex set V={1,2,…,n} and edge set E. The normalized Laplacian matrix of G is L=D−1/2(D−A)D−1/2, where D is the degree-diagonal matrix of G and A is the adjacency matrix of G. Let λ1≥λ2≥⋯≥λn−1≥λn=0 be the eigenvalues of L, for any real number α, the topological index sα∗(G) of G is defined as sα∗(G)=∑i=1n−1λiα.Let Ωij denote the resistance distance between vertices i and j, the degree-Kirchhoff index Kf∗(G) of G is defined as Kf∗(G)=∑i<jdidjΩij,where di is the degree of the vertex i. In this paper, we first express sα∗(G) in terms of Ωij’s explicitly for any real number α. Then as an application, we give a new proof for the classical Foster’s kth formula of resistance distances. Finally, we generalize the well-known relation 2|E|s−1∗(G)=Kf∗(G)to any integer k≥−1, which provides some upper bounds for sk∗(G).

Reconstruction of random geometric graphs: Breaking the [formula omitted] distortion barrier

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r=nα for any 0<α<1/2, with high probability reconstructs the vertex positions with a maximum error of O(nβ) where β=1/2−(4/3)α, until α≥3/8 where β=0 and the error becomes O(logn). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R3 and to hypercubes in any constant fixed dimension.

Chromatic number of a line with geometric progressions of forbidden distances and the complexity of recognizing distance graphs

Chromatic number of a line with geometric progressions of forbidden distances and the complexity of recognizing distance graphs

Modularity of some distance graphs

In this paper we study the modularity of G(n,r,s). We prove upper and lower bounds on the asymptotic behavior of this value for various values of r,s.

BNM-CDGNN: Batch Normalization Multilayer Perceptron Crystal Distance Graph Neural Network for Excellent-Performance Crystal Property Prediction.

Recently, in the field of crystal property prediction, the graph neural network (GNN) model has made rapid progress. The GNN model can effectively capture high-dimensional crystal features from the crystal structure, thereby achieving optimal performance in property prediction. However, the existing GNN model faces limitations in handling the hidden layer after the pooling layer, which restricts the training performance of the model. In the present research, we propose a novel GNN model called the batch normalization multilayer perceptron crystal distance graph neural network (BNM-CDGNN). BNM-CDGNN encodes the crystal's geometry structure only based on the distance vector between atoms. The graph convolutional layer utilizes the radial basis function as the attention mask, ensuring the crystal's rotation invariance and adding the geometric information on the crystal. Subsequently, the average pooling layer is connected after the convolutional layer to enhance the model's ability to learn precise information. BNM-CDGNN connects multiple hidden layers after the average pooling layers, and these layers are processed by the batch normalization layer. Finally, the fully connected layer maps the results to the target property. BNM-CDGNN significantly enhances the accuracy of crystal property prediction compared with previous baseline models such as SchNet, MPNN, CGCNN, MEGNet, and GATGNN.

Steiner Distance Related Parameters of Wheel graphs

For a connected graph of order at least and the Steiner distance among the vertices of is the minimum size among all the connected subgraphs whose vertex sets contain In this paper, we calculate the Steiner distance parameters such as Steiner degree distance, Steiner Gutman index, Steiner Harary index and Steiner reciprocal degree distance of a wheel graphs.

On the Span of ℓ Distance Coloring of Infinite Hexagonal Grid

For a graph [Formula: see text] and [Formula: see text], an [Formula: see text] distance coloring is a coloring [Formula: see text] of [Formula: see text] with [Formula: see text] colors such that [Formula: see text] when [Formula: see text]. Here [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] and is equal to the minimum number of edges that connect [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of [Formula: see text] distance coloring of [Formula: see text], [Formula: see text], is the minimum [Formula: see text] among all [Formula: see text] distance coloring of [Formula: see text]. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid [Formula: see text], and hence determining [Formula: see text] has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of [Formula: see text] for any odd [Formula: see text] and for even [Formula: see text], it is conjectured that [Formula: see text] where [Formula: see text] is an integer, [Formula: see text] and [Formula: see text]. For [Formula: see text], the conjecture has been proved by Ghosh and Koley [[Formula: see text]nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even [Formula: see text].

Square Distance in Graphs

In this paper, we consider a simple connected graph having no loops and multiple edges. The order and size of are denoted by and respectively in graphs is a wide branch of graph theory having many scientific and real-life applications. There are various types of distances studied in the literature.The distance is the length of the shortest path between the vertices and in [2]. The detour distance is the length of the longest path between the vertices and in [3]. The concept of centrality widely applied in social network analysis. Degree centrality is the simplest centrality measure to compute centre. Betweenness centrality measures used for analyzed cancer network.

Determining triangulations and quadrangulations by boundary distances

We show that if all internal vertices of a disc triangulation have degree at least 6, then the full structure can be determined from the pairwise graph distances between boundary vertices. A similar result holds for disc quadrangulations with all internal vertices having degree at least 4. This confirms a conjecture of Itai Benjamini. Both degree bounds are best possible, and correspond to local non-positive curvature. However, we show that a natural conjecture for a “mixed” version of the two results is not true.

Local intrinsic dimensionality measures for graphs, with applications to graph embeddings

The notion of local intrinsic dimensionality (LID) is an important advancement in data dimensionality analysis, with applications in data mining, machine learning and similarity search problems. Existing distance-based LID estimators were designed for tabular datasets encompassing data points represented as vectors in a Euclidean space. After discussing their limitations for graph-structured data considering graph embeddings and graph distances, we propose NC-LID, a novel LID-related measure for quantifying the discriminatory power of the shortest-path distance with respect to natural communities of nodes as their intrinsic localities. It is shown how this measure can be used to design LID-aware graph embedding algorithms by formulating two LID-elastic variants of node2vec with personalized hyperparameters that are adjusted according to NC-LID values. Our empirical analysis of NC-LID on a large number of real-world graphs shows that this measure is able to point to nodes with high link reconstruction errors in node2vec embeddings better than node centrality metrics. The experimental evaluation also shows that the proposed LID-elastic node2vec extensions improve node2vec by better preserving graph structure in generated embeddings.

Automorphism groups of the constituent graphs of integral distance graphs

In this paper, we consider the automorphism groups of Cayley graphs which are a basis of a complete Boolean algebra of strongly regular graphs, one of such graph is the integral distance graph Γ . The automorphism groups of the integral distance graphs Γ were determined by Kovács and Ruff in 2014 and by Kurz in 2009. Our point of interest will be restricted to the one determined by Kovács and Ruff. Here we consider the automorphism groups of subgraphs Γ 1 and Γ 2 , which inherit the properties of Γ . It will be shown that Aut Γ is isomorphic to Aut Γ 2 and that Aut Γ 1 is isomorphic to S F q 2 ≀ S [ 2 ] .

Dynamic Controllability of Temporal Plans in Uncertain and Partially Observable Environments

The formalism of Simple Temporal Networks (STNs) provides methods for evaluating the feasibility of temporal plans. The basic formalism deals with the consistency of quantitative temporal requirements on scheduled events. This implicitly assumes a single agent has full control over the timing of events. The extension of Simple Temporal Networks with Uncertainty (STNU) introduces uncertainty into the timing of some events. Two main approaches to the feasibility of STNUs involve (1) where a single schedule works irrespective of the duration outcomes, called Strong Controllability, and (2) whether a strategy exists to schedule future events based on the outcomes of past events, called Dynamic Controllability. Case (1) essentially assumes the timing of uncertain events cannot be observed by the agent while case (2) assumes full observability. The formalism of Partially Observable Simple Temporal Networks with Uncertainty (POSTNU) provides an intermediate stance between these two extremes, where a known subset of the uncertain events can be observed when they occur. A sound and complete polynomial algorithm to determining the Dynamic Controllability of POSTNUs has not previously been known; we present one in this paper. This answers an open problem that has been posed in the literature. The approach we take factors the problem into Strong Controllability micro-problems in an overall Dynamic Controllability macro-problem framework. It generalizes the notion of labeled distance graph from STNUs. The generalized labels are expressed as max/min expressions involving the observables. The paper introduces sound generalized reduction rules that act on the generalized labels. These incorporate tightenings based on observability that preserve dynamic viable strategies. It is shown that if the generalized reduction rules reach quiescence without exposing an inconsistency, then the POSTNU is Dynamically Controllable (DC). The paper also presents algorithms that apply the reduction rules in an organized way and reach quiescence in a polynomial number of steps if the POSTNU is Dynamically Controllable. Remarkably, the generalized perspective leads to a simpler and more uniform framework that applies also to the STNU special case. It helps illuminate the previous methods inasmuch as the max/min label representation is more semantically clear than the ad-hoc upper/lower case labels previously used.

Coloring the distance graphs

Coloring the distance graphs

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an m×r integer matrix, where we call m the dimension and r the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. As a demonstration of its utility, we provide an alternate proof of Payan's theorem, which states that a cubelike graph (i.e., a Cayley graph on the product Z2×⋯×Z2 of the integers modulo 2 with itself finitely many times) cannot have chromatic number 3. In a series of follow-up articles using the method of Heuberger matrices, we completely determine the chromatic number in cases with small dimension and rank, as well as prove a generalization of Zhu's theorem on the chromatic number of 6-valent integer distance graphs.

On Some Non-Rigid Unit Distance Patterns

A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius 1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/{\\sqrt{2}}$$\\end{document}. We also consider a similar problem about 3-regular unit distance graphs in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Estimating distance between an eigenvalue of a signed graph and the spectrum of an induced subgraph

The distance between an eigenvalue λ of a signed graph Ġ and the spectrum of a signed graph Ḣ is defined as min{|λ−μ|:μis an eigenvalue ofḢ}. In this paper, we investigate this distance when Ḣ is a largest induced subgraph of Ġ that does not have λ as an eigenvalue. We estimate the distance in terms of eigenvectors and structural parameters related to vertex degrees. For example, we show that |λ||λ−μ|≤δĠ∖Ḣmax{dḢ(i)dĠ−V(Ḣ)(j):i∈V(Ġ)∖V(Ḣ),j∈V(Ḣ),i∼j}, where δĠ∖Ḣ is the minimum vertex degree in V(Ġ)∖V(Ḣ). If Ḣ is obtained by deleting a single vertex i, this bound reduces to |λ||λ−μ|≤d(i). We also consider the case in which λ is a simple eigenvalue and Ḣ is not necessarily a vertex-deleted subgraph, and the case when λ is the largest eigenvalue of an ordinary (unsigned) graph. Our results for signed graphs apply to ordinary graphs. They can be interesting in the context of eigenvalue distribution, eigenvalue location or spectral distances of (signed) graphs.

On reciprocal degree distance of graphs

Given a connected graph H, its reciprocal degree distance is defined asRDD(H)=∑x≠ydH(vx)+dH(vy)dH(vx,vy), where dH(vx) denotes the degree of the vertex vx in the graph H and dH(vx,vy) is the shortest distance between vx and vy in H. The goal of this paper is to establish some sufficient conditions to judge that a graph to be ħ-hamiltonian, ħ-path-coverable or ħ-edge-hamiltonian by employing the reciprocal degree distance.

An efficient graph‐based peer selection method for financial statements

SummaryComparing companies can be useful for various purposes. Despite the widespread use of industry classification systems as a peer selection standard, these have been criticized for various reasons. Financial statements, however, offer a promising alternative to such classification systems. They are standardized, widely available, and offer deep insights into the nature of the company. In this paper, we present a graph distance metric for financial statements using the earth mover's distance. When using the distance metric on real‐world tasks such as peer identification and industry classification, it shows promising results in terms of accuracy and computational efficiency.

Study State Dynamics of Team Passing Networks in Soccer Games

ABSTRACT Complex networks have been widely used in studying collective behaviours in soccer sports, such as examining tactical strategies, recognizing team characteristics, and discovering topological determinants for high team performance. The passing network of a team evolves and displays different temporal patterns, that are strongly linked to team status, tactical strategies, attacking/defending transitions, etc. Nevertheless, existing research has not illuminated the state dynamics of team passing networks, whereas similar methods have been extensively used in examining the dynamical brain networks constructed from human brain neuroimage data. This study aims to investigate the state dynamics of team passing networks in soccer sports. The introduced method incorporates multiple techniques, including sliding time window, network modeling, graph distance measure, clustering, and cluster validation. The final match of the FIFA World Cup 2018 was taken as an example, and the state dynamics of teams Croatia and France were analyzed respectively. Additionally, the effects of the time windows and graph distance measures on the results were briefly discussed. This study presents a novel outlook on examining the dynamics of team passing networks, as it facilitates the recognition of important team states or state transitions in soccer and other team ball-passing sports for further analysis.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.