Graph Distance Research Articles

Spectral deviations of graphs

Abstract For the graphs G G and H H , the spectral deviation of H H from G G is defined as ϱ G ( H ) = ∑ μ ∈ H min λ ∈ G ∣ λ − μ ∣ , {\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈ \in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph H H of a prescribed graph G G such that ϱ G ( H ) = 0 {\varrho }_{G}\left(H)=0 , and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs.

Resistance distances in generalized join graphs

Resistance distances in generalized join graphs

First Passage Percolation, Local Uniqueness for Interlacements and Capacity of Random Walk

The study of first passage percolation (FPP) for the random interlacements model has been initiated in Andres and Prévost (Ann Appl Probab 34(2):1846–1895), where it is shown that on Zd, d≥3, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.

A Novel Spectral Graph Distance Measure and Its Applications in Biomedical Signal Processing

A Novel Spectral Graph Distance Measure and Its Applications in Biomedical Signal Processing

Connectome-based prediction of functional impairment in experimental stroke models.

Experimental rat models of stroke and hemorrhage are important tools to investigate cerebrovascular disease pathophysiology mechanisms, yet how significant patterns of functional impairment induced in various models of stroke are related to changes in connectivity at the level of neuronal populations and mesoscopic parcellations of rat brains remain unresolved. To address this gap in knowledge, we employed two middle cerebral artery occlusion models and one intracerebral hemorrhage model with variant extent and location of neuronal dysfunction. Motor and spatial memory function was assessed and the level of hippocampal activation via Fos immunohistochemistry. Contribution of connectivity change to functional impairment was analyzed for connection similarities, graph distances and spatial distances as well as the importance of regions in terms of network architecture based on the neuroVIISAS rat connectome. We found that functional impairment correlated with not only the extent but also the locations of the injury among the models. In addition, via coactivation analysis in dynamic rat brain models, we found that lesioned regions led to stronger coactivations with motor function and spatial learning regions than with other unaffected regions of the connectome. Dynamic modeling with the weighted bilateral connectome detected changes in signal propagation in the remote hippocampus in all 3 stroke types, predicting the extent of hippocampal hypoactivation and impairment in spatial learning and memory function. Our study provides a comprehensive analytical framework in predictive identification of remote regions not directly altered by stroke events and their functional implication.

Dynamic automated assessment of functionally limited temporomandibular disorders using motion analysis.

This study investigated a method for assessing movement patterns in patients with functionally limited temporomandibular disorders using minimal equipment. The aim was to enable clinicians to better understand temporomandibular joint function, support treatment decision-making, and provide data for documenting disease progression. This retrospective, observational study included 45 patients with functionally limited temporomandibular disorders (Group A) and 40 healthy volunteers (Group B). Video data of temporomandibular joint movements were recorded. Measurements, including maximum mouth opening distance, maximum offset distance, maximum offset angle, direction of offset, and fluctuation graphs of offset angle and distance, were obtained using manual methods and an automatic measurement program. The results were statistically analyzed. Statistically significant differences were observed between patients and the control group in the manually measured mouth opening distance and offset distance, as well as in the automatically measured offset distance and offset angle. Automated testing methods can effectively assess the motor function of functionally limited temporomandibular joints, providing more objective, data-driven, and graphical testing indexes. Further research is warranted to explore their practical significance.

Bootstrap percolation, connectivity, and graph distance

Bootstrap percolation, connectivity, and graph distance

An embedding-based distance for temporal graphs.

Temporal graphs are commonly used to represent time-resolved relations between entities in many natural and artificial systems. Many techniques were devised to investigate the evolution of temporal graphs by comparing their state at different time points. However, quantifying the similarity between temporal graphs as a whole is an open problem. Here, we use embeddings based on time-respecting random walks to introduce a new notion of distance between temporal graphs. This distance is well-defined for pairs of temporal graphs with different numbers of nodes and different time spans. We study the case of a matched pair of graphs, when a known relation exists between their nodes, and the case of unmatched graphs, when such a relation is unavailable and the graphs may be of different sizes. We use empirical and synthetic temporal network data to show that the distance we introduce discriminates graphs with different topological and temporal properties. We provide an efficient implementation of the distance computation suitable for large-scale temporal graphs.

Wind farm cluster power prediction based on graph deviation attention network with learnable graph structure and dynamic error correction during load peak and valley periods

Wind farm cluster power prediction based on graph deviation attention network with learnable graph structure and dynamic error correction during load peak and valley periods

Sparse graph structure fusion convolutional network for machinery remaining useful life prediction

Sparse graph structure fusion convolutional network for machinery remaining useful life prediction

Steiner 3-Wiener Index of Zigzag Polyhex Nanotubes.

Let G be a connected graph and S be a k element subset of the vertex set V(G) of G. Steiner distance is a natural generalization of the usual graph distance. The Steiner-k distance dG(S) between the vertices of S is the minimum size among all connected subgraphs whose vertex set contains S. The generalized indices based on Steiner distances have several applications in the real world. It is a well-known fact that "Steiner Problem" is NP-complete and hence any parameter based on Steiner distance is also an NP problem. The objective of this work is to determine the Steiner 3-Wiener index for an important chemical structure called the zigzag polyhex nanotube. In this paper, we present an algorithm for computing the Steiner 3-Wiener index (SW3) for zigzag polyhex nanotubes. The developed algorithm addresses the complexities associated with exponential increments in the number of vertices as the nanotube's circumference or length expands. The obtained SW3 values for zigzag polyhex nanotubes can be used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) analyses. We have presented an algorithm and listed the numerical values of SW3 for various parameters of circumference and length of a zigzag polyhex nanotube to facilitate their utilization in QSAR/ QSPR analyses. We have obtained the time complexity for the algorithm, which shows that the SW3 values are computationally intensive. To explain this complex nature, we have used multiple linear regression to fit log SW3 values corresponding to log p and log q, the radius and length of a nanotube. The algorithm addresses the complexities associated with the exponential increase in vertices as the nanotube's circumference or length expands. Furthermore, we provide SW3 values for various parameter combinations up to circumference or length 17, and a general relation to determine the value of SW3, facilitating its utilization in real-world applications. These values serve as crucial descriptors for understanding the structural nuances of zigzag polyhex nanotubes, that find applications in material science, drug design, drug discovery, etc.

Chromatic Coloring of Distance Graphs V

Let G = ( V , E ) be any graph. If there exists an injection f : V → Z , such that | f ( u ) – f ( v ) | is prime for every u v ∈ E , then we say G is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of n mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of n arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

Predictive potential of distance-related spectral graphical descriptors for structure-property modeling of thermodynamic properties of polycyclic hydrocarbons with applications

A distance-related spectral descriptor is a graphical index with defining structure built on eigenvalues of chemical matrices relying on distances in graphs. This paper explores the predictive ability of both existing and new distance-related spectral descriptors for estimating thermodynamic characteristics of polycyclic hydrocarbons (PHs). As a standard choice, the entropy and heat capacity are selected to represent thermodynamic properties. Furthermore, 30 initial members of PHs are considered as test molecules for this study. Three new molecular matrices have been proposed and our research demonstrates that distance-spectral graphical indices built by these novel matrices surpass in efficiency relative to famous distance-spectral indices. First, a novel computational method is put forwarded to evaluate distance-spectral indices of molecular graphs. The proposed methodology is utilized to compute both pre-existing and novel distance-related spectral descriptors, with an aim to assess their predictive efficacy using experimental data pertaining to two selected thermodynamic properties. Subsequently, we identify the five most promising distance-related spectral descriptors, comprising the degree-distance and Harary energies, the recently introduced second geometric-arithmetic energy along with its associated Estrada invariant, and 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {nd}$$\\end{document} atom-bond connectivity (ABC) Estrada index. Notably, the 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {nd}$$\\end{document} ABC Estrada index and Harary energy demonstrate correlation coefficients exceeding 0.95, while certain conventional spectral indices including the distance energy as well as its associated Estrada index, display comparatively lower performance levels. Moreover, we illustrate the practical implications of our findings on specific classes of one-hexagonal nanocones and carbon polyhex nanotubes. These outcomes hold potential for enhancing the theoretical determination of certain thermodynamic attributes of these nanostructures, offering improved accuracy and minimal margin of error.

Connected 𝐷 - Eccentric Domination in Graphs

Objectives: To introduce connected -eccentric point set, connected -eccentric number, connected -eccentric dominating set, connected -eccentric domination number in a graph and related concepts. Methods: -distance in graphs are used to find the connected -eccentric number and connected -eccentric domination number in graphs. Findings: The new term connected -eccentric domination in graphs are used in varies field like to construct least number of cell phone tower in low cost and traffic signal also. Novelty: Using the idea -distance, eccentricity in a graphs, the connected -eccentric dominating set and its number in graphs are obtained. Some points, observation, bounds and theorem related to connected -eccentric domination set and its number in graphs are stated and proved. 2010 Mathematics Subject Classification: 05C12, 05C69 Keywords: D-eccentric node, D-eccentric node-set, D-eccentric dominating set, D-eccentric domination number, Connected D-eccentric dominating set, Connected D-eccentric domination number

Auto-weighted Graph Reconstruction for efficient ensemble clustering

Auto-weighted Graph Reconstruction for efficient ensemble clustering

Labeled interleaving distance for Reeb graphs

Labeled interleaving distance for Reeb graphs

A Bellman–Ford Algorithm for the Path-Length-Weighted Distance in Graphs

A Bellman–Ford Algorithm for the Path-Length-Weighted Distance in Graphs

Building shape-focused pharmacophore models for effective docking screening

The performance of molecular docking can be improved by comparing the shape similarity of the flexibly sampled poses against the target proteins’ inverted binding cavities. The effectiveness of these pseudo-ligands or negative image-based models in docking rescoring is boosted further by performing enrichment-driven optimization. Here, we introduce a novel shape-focused pharmacophore modeling algorithm O-LAP that generates a new class of cavity-filling models by clumping together overlapping atomic content via pairwise distance graph clustering. Top-ranked poses of flexibly docked active ligands were used as the modeling input and multiple alternative clustering settings were benchmark-tested thoroughly with five demanding drug targets using random training/test divisions. In docking rescoring, the O-LAP modeling typically improved massively on the default docking enrichment; furthermore, the results indicate that the clustered models work well in rigid docking. The C+ +/Qt5-based algorithm O-LAP is released under the GNU General Public License v3.0 via GitHub (https://github.com/jvlehtonen/overlap-toolkit).Scientific contributionThis study introduces O-LAP, a C++/Qt5-based graph clustering software for generating new type of shape-focused pharmacophore models. In the O-LAP modeling, the target protein cavity is filled with flexibly docked active ligands, the overlapping ligand atoms are clustered, and the shape/electrostatic potential of the resulting model is compared against the flexibly sampled molecular docking poses. The O-LAP modeling is shown to ensure high enrichment in both docking rescoring and rigid docking based on comprehensive benchmark-testing.Graphical

Graph Numbers and Distance Related Parameters of Zero Divisor Graphs

Graph Numbers and Distance Related Parameters of Zero Divisor Graphs

K-Means Clustering With Natural Density Peaks for Discovering Arbitrary-Shaped Clusters.

Due to simplicity, K-means has become a widely used clustering method. However, its clustering result is seriously affected by the initial centers and the allocation strategy makes it hard to identify manifold clusters. Many improved K-means are proposed to accelerate it and improve the quality of initialize cluster centers, but few researchers pay attention to the shortcoming of K-means in discovering arbitrary-shaped clusters. Using graph distance (GD) to measure the dissimilarity between objects is a good way to solve this problem, but computing the GD is time-consuming. Inspired by the idea that granular ball uses a ball to represent the local data, we select representatives from a local neighborhood, called natural density peaks (NDPs). On the basis of NDPs, we propose a novel K-means algorithm for identifying arbitrary-shaped clusters, called NDP-Kmeans. It defines neighbor-based distance between NDPs and takes advantage of the neighbor-based distance to compute the GD between NDPs. Afterward, an improved K-means with high-quality initial centers and GD is used to cluster NDPs. Finally, each remaining object is assigned according to its representative. The experimental results show that our algorithms can not only recognize spherical clusters but also manifold clusters. Therefore, NDP-Kmeans has more advantages in detecting arbitrary-shaped clusters than other excellent algorithms.