Linear Graph Research Articles

One-to-one Disjoint-path Covers of Leaf-sort Graphs

Parallel paths of interconnection networks have attracted much attention in parallel computing systems, they are studied in terms of disjoint paths in an undirected graph [Formula: see text]. A [Formula: see text]-disjoint path cover (k-DPC for short) of a graph [Formula: see text] is a set of [Formula: see text] (internally) disjoint paths that altogether cover every vertice of the graph. A bipartite graph [Formula: see text] is one-to-one bi-k-DPC if there is a k-DPC between any pair of vertices in different partite sets. A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any pair of vertices in different partite sets. The [Formula: see text]-dimensional leaf-sort graph [Formula: see text] has many good properties, including bipartite, symmetry, vertex-transitive. In this paper, we prove that [Formula: see text] has one-to-one bi-[Formula: see text]-DPC between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] in different partite sets, and [Formula: see text] is hamiltonian laceable.

Standardization of color measurement in the medical photography in clinical practice

BACKGROUND: For many medical specialties, photography is a routine element of work. Environmental factors equally and significantly influence the interpretation of colour light perception and the quality of photographic recording. Standardization of conditions in medical photography is necessary for subsequent quantitative assessment of the postoperative skin scar by analysing colour coordinates in the RGB system. AIM: To determine the impact of illumination and shooting distance at colour coordinates when studying unaltered skin. METHODS: Five volunteers without any skin diseases and no make-up on their facial skin were taken in the study. Skin type by the Fitzpatrick scale was II–III. Digital photography was made with a mobile device camera at distance 30 cm from the subject. For each participant, a series of digital photographs was taken at a distance of 20, 30, 40, 50, 60, 70 cm from the light source. Colour coordinates in the RGB system were measured in the digital graphics editor Adobe Photoshop CS6. Statistical analysis of the obtained data was carried out using Microsoft Office Excel 2019, Phyton 3.11. RESULTS: The analysis of variance by the ANOVA method was used as a statistical analysis. In order to determine a statistically significant difference between the sets, Tukey HSD test was performed. A total of 1764 coordinates of three colours were subjected to statistical analysis (R=588, G=588, B=588). When constructing a heat map of the cross-correlation of the absolute values of each colour at each distance, taking into account the type of light source among themselves, the data had a strong direct correlation, regardless of the study area. When constructing a linear graph, coordinates of any of the zones were located on the same straight line. CONCLUSION: Photography conditions were experimentally determined under which the color interpretation of light is constant. Results of the study should be taken into account during medical photography and subsequent color assessment of postoperative skin scars.

Structural robustness and optimization of leader-follower coherence in star-coupled networks

Abstract In this paper, we investigate the structural robustness and optimization of leader-follower coherence, quantified by the eigenvalues of the grounded Laplacian matrix, which measures the deviation between leaders and followers. To examine the impacts of network couplings and leader assignments on coherence, we select star-coupled networks as base models and introduce three types of coupling forms. By utilizing regular network structures, we derive analytical expressions for leader-follower coherence with respect to network parameters and evaluate the performance of coherence under various leader assignments. In addition, for achieving the lowest coherence in a network connected by a path graph, we propose an optimization algorithm that involves adding edges among the followers to enhance coherence.

Edge‐arc‐disjoint paths in semicomplete mixed graphs

AbstractThe so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

Fast Algorithm for Cyber-Attack Estimation and Attack Path Extraction Using Attack Graphs with AND/OR Nodes

This paper studies the security issues for cyber–physical systems, aimed at countering potential malicious cyber-attacks. The main focus is on solving the problem of extracting the most vulnerable attack path in a known attack graph, where an attack path is a sequence of steps that an attacker can take to compromise the underlying network. Determining an attacker’s possible attack path is critical to cyber defenders as it helps identify threats, harden the network, and thwart attacker’s intentions. We formulate this problem as a path-finding optimization problem with logical constraints represented by AND and OR nodes. We propose a new Dijkstra-type algorithm that combines elements from Dijkstra’s shortest path algorithm and the critical path method. Although the path extraction problem is generally NP-hard, for the studied special case, the proposed algorithm determines the optimal attack path in polynomial time, O(nm), where n is the number of nodes and m is the number of edges in the attack graph. To our knowledge this is the first exact polynomial algorithm that can solve the path extraction problem for different attack graphs, both cycle-containing and cycle-free. Computational experiments with real and synthetic data have shown that the proposed algorithm consistently and quickly finds optimal solutions to the problem.

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

Zagreb Polynomials, Co-Polynomials of Bipartite and Strongly Bipartite Graphs

First Zagreb polynomial of a graph G with vertex set V(G) and edge set E(G) is defined as M1(G,x) =∑uv∈E(G)xdu+dv and the first Zagreb index can be obtained from it as M1(G) = M1(G,x)|x=1. In this paper Zagreb polynomials, co-polynomials and corresponding Zagreb indices, co-indices are obtained for path, cycle, double and strong double graphs.

Geometric deep learning framework for de novo genome assembly.

The critical stage of every de novo genome assembler is identifying paths in assembly graphs that correspond to the reconstructed genomic sequences. The existing algorithmic methods struggle with this, primarily due to repetitive regions causing complex graph tangles, leading to fragmented assemblies. Here, we introduce GNNome, a framework for path identification based on geometric deep learning that enables training models on assembly graphs without relying on existing assembly strategies. By leveraging only the symmetries inherent to the problem, GNNome reconstructs assemblies from PacBio HiFi reads with contiguity and quality comparable to those of the state-of-the-art tools across several species. With every new genome assembled telomere-to-telomere, the amount of reliable training data at our disposal increases. Combining the straightforward generation of abundant simulated data for diverse genomic structures with the AI approach makes the proposed framework a plausible cornerstone for future work on reconstructing complex genomes with different ploidy and aneuploidy degrees. To facilitate such developments, we make the framework and the best-performing model publicly available, provided as a tool that can directly be used to assemble new haploid genomes.

Prevalence Trends of Foodborne Pathogens Bacillus cereus, Non-STEC Escherichia coli and Staphylococcus aureus in Ready-to-Eat Foods Sourced from Restaurants, Cafés, Catering and Takeaway Food Premises

Foodborne pathogens of Bacillus cereus (B. cereus), non-STEC Escherichia coli (non-STEC E. coli) and Staphylococcus aureus (S. aureus) are currently non-notifiable in Australia unless attributed to a food poisoning outbreak. Due to the lack of data around individual cases and isolations in foods, any changes in prevalence may go undetected. The aim of this study was to determine any changes in the prevalence of B. cereus, non-STEC E. coli and S. aureus in ready-to-eat (RTE) foods sampled from Western Australian restaurants, cafés, catering facilities and takeaway food premises from July 2009 to June 2022. A total of 21,822 microbiological test results from 7329 food samples analysed over this 13-year period were reviewed and analysed. Linear trend graphs derived from the annual prevalence and binary logistic regression models were used to analyse the sample results, which indicated an increase in prevalence for B. cereus. In contrast, a decrease in prevalence for both S. aureus and non-STEC E. coli was determined. Additionally, there were changes in prevalence for the three bacteria in specific months, seasons, specific RTE foods and food premises types. Further research is needed to gain a better understanding of the potential drivers behind these changes in prevalence, including the potential impacts of climate change, COVID-19, legislation and guidelines targeting specific RTE foods, and the difficulty of differentiating B. cereus from B. thuringeniesis using standard testing methods.

GTasm: a genome assembly method using graph transformers and HiFi reads.

Genome assembly aims to reconstruct the whole chromosome-scale genome sequence. Obtaining accurate and complete chromosome-scale genome sequence serve as an indispensable foundation for downstream genomics analyses. Due to the complex repeat regions contained in genome sequence, the assembly results commonly are fragmented. Long reads with high accuracy rate can greatly enhance the integrity of genome assembly results. Here we introduce GTasm, an assembly method that uses graph transformer network to find optimal assembly results based on assembly graphs. Based on assembly graph, GTasm first extracts features about vertices and edges. Then, GTasm scores the edges by graph transformer model, and adopt a heuristic algorithm to find optimal paths in the assembly graph, each path corresponding to a contig. The graph transformer model is trained using simulated HiFi reads from CHM13, and GTasm is compared with other assembly methods using real HIFI read set. Through experimental result, GTasm can produce well assembly results, and achieve good performance on NA50 and NGA50 evaluation indicators. Applying deep learning models to genome assembly can improve the continuity and accuracy of assembly results. The code is available from https://github.com/chu-xuezhe/GTasm.

An optimized siamese neural network with deep linear graph attention model for gynaecological abdominal pelvic masses classification.

An adnexal mass, also known as a pelvic mass, is a growth that develops in or near the uterus, ovaries, fallopian tubes, and supporting tissues. For women suspected of having ovarian cancer, timely and accurate detection of a malignant pelvic mass is crucial for effective triage, referral, and follow-up therapy. While various deep learning techniques have been proposed for identifying pelvic masses, current methods are often not accurate enough and can be computationally intensive. To address these issues, this manuscript introduces an optimized Siamese circle-inspired neural network with deep linear graph attention (SCINN-DLGN) model designed for pelvic mass classification. The SCINN-DLGN model is intended to classify pelvic masses into three categories: benign, malignant, and healthy. Initially, real-time MRI pelvic mass images undergo pre-processing using semantic-aware structure-preserving median morpho-filtering to enhance image quality. Following this, the region of interest (ROI) within the pelvic mass images is segmented using an EfficientNet-based U-Net framework, which reduces noise and improves the accuracy of segmentation. The segmented images are then analysed using the SCINN-DLGN model, which extracts geometric features from the ROI. These features are classified into benign, malignant, or healthy categories using a deep clustering algorithm integrated into the linear graph attention model. The proposed system is implemented on a Python platform, and its performance is evaluated using real-time MRI pelvic mass datasets. The SCINN-DLGN model achieves an impressive 99.9% accuracy and 99.8% recall, demonstrating superior efficiency compared to existing methods and highlighting its potential for further advancement in the field.

Exploring Mean Cordial Labeling Possible Combination Position of Vertices in Graphs: A Computational Approach

Objectives: We have developed the Python module of mean cordial labeling for the different types of graphs, like Path, Cycle, and Subdivision of Star graphs. We aim to identify the number of combination positions of vertices that satisfy the mean cordial labeling rules. We have found mathematical equations that describe the labeling behavior in these graphs. Methods: In this paper, a Python program was developed to find the mean cordial labeling of the Path, Cycle, and Subdivision of the Star graph. We have obtained the number of combination positions of vertex obeying the mean cordial labeling condition using this Python module. We have drawn the graph between the number of vertices and the number of combination positions of the vertex. Findings: We have found that the different graph types like Path, Cycle, and Subdivision of Star graph satisfy the mean cordial labeling condition. The result indicates that the number of combination positions of the Path and cycle graph fulfill the power equation Y=a*xb and the Subdivision of the Star graph meets the exponential growth equation Y=a*bx. Novelty: In this paper, a new method has been developed to find the number of combination position of varies graphs using the Python module. This computational approach will be useful to analyze the biological networking and protein-protein interaction study. Keywords: Mean Cordial, Path, Cycle, Subdivision of Star graph, Computational graph theory

Polynomial Time Algorithm for Shortest Paths in Interval Temporal Graphs

We develop a polynomial time algorithm for the single-source all destinations shortest paths problem for interval temporal graphs (ITGs). While a polynomial time algorithm for this problem is known for contact sequence temporal graphs (CSGs), no such prior algorithm is known for ITGs. We benchmark our ITG algorithm against that for CSGs using datasets that can be solved using either algorithm. Using synthetic datasets, experimentally, we show that our algorithm for ITGs obtains a speedup of up to 32.5 relative to the state-of-the-art algorithm for CSGs.

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

Knowledge-Enhanced Conversational Recommendation via Transformer-Based Sequential Modeling

In conversational recommender systems (CRSs), conversations usually involve a set of items and item-related entities or attributes, e.g., director is a related entity of a movie. These items and item-related entities are often mentioned along the development of a dialog, leading to potential sequential dependencies among them. However, most of existing CRSs neglect these potential sequential dependencies. In this article, we first propose a Transformer-based sequential conversational recommendation method, named TSCR, to model the sequential dependencies in the conversations to improve CRS. In TSCR, we represent conversations by items and the item-related entities, and construct user sequences to discover user preferences by considering both the mentioned items and item-related entities. Based on the constructed sequences, we deploy a Cloze task to predict the recommended items along a sequence. Meanwhile, in certain domains, knowledge graphs formed by the items and their related entities are readily available, which provide various different kinds of associations among them. Given that TSCR does not benefit from such knowledge graphs, we then propose a knowledge graph enhanced version of TSCR, called TSCRKG. In specific, we leverage the knowledge graph to offline initialize our model TSCRKG, and augment the user sequence of conversations (i.e., sequence of the mentioned items and item-related entities in the conversation) with multi-hop paths in the knowledge graph. Experimental results demonstrate that our TSCR model significantly outperforms state-of-the-art baselines, and the enhanced version TSCRKG further improves recommendation performance on top of TSCR.

Upper Bounds for Linear Graph Codes

ABSTRACTA linear graph code is a family of graphs on vertices with the property that the symmetric difference of the edge sets of any two graphs in is also the edge set of a graph in . In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph . In particular, we show that if has an even number of edges, the size of the code is , making progress on a question of Alon. Furthermore, we show that for almost all graphs with an even number of edges, there exists such that the size of a linear graph code without a copy of is at most .

On the Energy Behaviors of the Bellman–Ford and Dijkstra Algorithms: A Detailed Empirical Study

The Single-Source Shortest Paths (SSSP) graph problem is a fundamental computation. This study attempted to characterize concretely the energy behaviors of the two primary methods to solve it, the Bellman–Ford and Dijkstra algorithms. The very different interactions of the algorithms with the hardware may have significant implications for energy. The study was motivated by the multidisciplinary nature of the problem. Gaining better insights should help vital applications in many domains. The work used reliable embedded sensors in an HPC-class CPU to collect empirical data for a wide range of sizes for two graph cases: complete as an upper-bound case and moderately dense. The findings confirmed that Dijkstra’s algorithm is drastically more energy efficient, as expected from its decisive time complexity advantage. In terms of power draw, however, Bellman–Ford had an advantage for sizes that fit in the upper parts of the memory hierarchy (up to 2.36 W on average), with a region of near parity in both power draw and total energy budgets. This result correlated with the interaction of lighter logic and graph footprint in memory with the Level 2 cache. It should be significant for applications that rely on solving a lot of small instances since Bellman–Ford is more general and is easier to implement. It also suggests implications for the design and parallelization of the algorithms when efficiency in power draw is in mind.

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^{k}n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman on finding a path through \(k\) specified vertices. It also implies the first \(2^{k}n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) . Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T\subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^{2}\log(q+k))}n^{\mathcal{O}(1)}w\) time randomized algorithm for this problem.

Research on Evaluation Method of Natural Gas Hydrate Resources Based on Probability Distribution Model

Natural gas hydrate, as a highly efficient and clean energy source, has garnered significant attention from both developed and energy-deficient nations. However, to achieve industrial application, several technical challenges must be addressed, including resource exploration, spatial positioning, resource assessment, economic production evaluation, and climate change impact assessment. Currently, our exploration and evaluation system for gas hydrate resources remains immature. This study aims to determine the probability distribution of effective thickness, porosity, and saturation of natural gas hydrate resource parameters in the study area and their variations. Utilizing probability statistics theory, we calculated specific values for all gas hydrate resource parameters from open-source data of 14 exploration well locations, determined their probability distributions, and conducted distribution detection to identify specific distributions. Data analysis was performed using linear graphs and bar charts to summarize patterns. Subsequently, by applying the mathematical formula Q = A×Z×∅×S×E, we categorized natural gas hydrate resource parameters, calculated gas hydrate resources at each effective depth, determined the probability distribution of each obtained data point, and ultimately solved for the total natural gas hydrate resources, yielding a final solution of 31875763695.08.

Rainbow Mean Index of Some Classes of Graphs

In this article, we determine the rainbow mean index of cartesianproductG◻Pn, where G is a regular graph;Pm◻Pn; Wn◻K2;total graph of paths; chain of cycles and complete graphs; triangular treeand join graphs tCs∨K1.