Combinatorial optimization of special graphs for nodal ordering and graph partitioning

Root systems and inflations of non-negative quasi-Cartan matrices

PurposeThe purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.Design/methodology/approachIn the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.FindingsA new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.Originality/valueThe present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

A multi-level finite element nodal ordering using algebraic graph theory

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

The m‐way graph partitioning problem is of central importance in combinatorial optimization. It has many important applications in fields such as VLSI circuit design, task allocation in distributed computing systems, and network partitioning. In this paper, we propose an efficient genetic algorithm to solve this problem. The proposed method searches a large solution space and finds the best possible solution by adjusting the intensification and diversification automatically during the optimization process. The proposed method is tested on a large number of instances and compared with some existing algorithms. The experimental results show that the proposed algorithm is superior to its competitors in terms of computation time and solution quality. © 2011 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Graph representation learning (GRL) aims to map a graph into a low-dimensional vector space while preserving graph topology and node properties. This study proposes a novel GRL model, Multi-Level GRL (simply, ML-GRL), that recursively partitions input graphs by selecting the most appropriate community detection algorithm at each graph or partitioned subgraph. To preserve the relationship between subgraphs, ML-GRL incorporates global graphs that effectively maintain the overall topology. ML-GRL employs a prediction model, which is pre-trained using graph-based features and covers a wide range of graph distributions, to estimate GRL accuracy of each community detection algorithm without partitioning graphs or subgraphs and evaluating them. ML-GRL improves learning accuracy by selecting the most effective community detection algorithm while enhancing learning efficiency from parallel processing of partitioned subgraphs. Through extensive experiments with two different tasks, we demonstrate ML-GRL's superiority over the six representative GRL models in terms of both learning accuracy and efficiency. Specifically, ML-GRL not only improves the accuracy of existing GRL models by 3.68 ~ 47.59% for link prediction and 1.75 ~ 40.90% for node classification but also significantly reduces their running time by 9.63 ~ 62.71% and 7.14 ~ 82.14%, respectively. Our source code is available at https://github.com/pnpy6elp/Multi_Level_GRL.

Contributors

Topological indices are numbers that are applied to a graph and can be used to describe specific graph properties through algebraic structures. Algebraic graph theory is a helpful tool in a range of chemistry domains. Because it helps explain how the different symmetries of molecules and crystals affect their structure and dynamics, it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. A topological index converts the chemical structure into a number and contributes a lot in chemical graph theory. In this article, we compute the Wiener index, Zagreb indexes, Wiener polynomial, Hyper-Wiener index, ABC index and eccentricity-based topological index of a nonzero component union graph from vector space.

Graph products for configuration processing of space structures

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it is shown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

Flocking of a team of Lagrangian agents is investigated in this paper. The motion equation of agents with Lagrangian dynamics and its properties are reviewed. The centrifugal and coriolis effects and gravitational terms in the dynamics of agents have been neglected in previous works on flocking problem. However the dynamics of many dynamical systems which would be the agents in practical applications can be assumed as Lagrangian with the above mentioned terms. The proposed flocking scheme is based on a gradient algorithm in which the collective potential is systematically constructed to cause the velocity of the agents to reach a consensus. It is explained that the suggested algorithm satisfies three rules of flocking demonstrated by Reynolds: cohesion, separation, and alignment. For this purpose the Hamiltonian of a team of agents with Lagrangian dynamics is considered and is treated as the Lyapunov like for the Multi-Agent system. Then considering the properties of the Lagrangian systems and also using the algebraic graph theory, it is shown that the derivative of the Hamiltonian is in order that the Reynolds' rules are satisfied and the conditions under which flocking is achieved are derived. The analytical tools provided here are relayed on algebraic graph theory, matrix theory, and control theory. In simulation results, a group of wheeled mobile robots are used to show the effectiveness of the suggested algorithm.

The shortest route finding problem is a significant topic in graph theory and combinatorial optimization, with wide applications in logistics, transportation, and scheduling. This research aims to improve the quality of the solution and time efficiency in solving the Traveling Salesman Problem (TSP) by optimizing the Cheapest Insertion Heuristic (CIH) algorithm using the application of the Tabu Search algorithm. The CIH algorithm constructs an initial solution by inserting points based on minimum weight. At the same time, the Tabu Search algorithm is applied to enhance the solution by avoiding local optima using a tabu list mechanism. The research data, consisting of the distances between parking retribution collection points by the Malang City Transportation Agency in Sukun Sub-district, were obtained from Google Maps. The algorithm performance evaluation is done by comparing the total mileage before and after optimization, and statistically analyzed using the Wilcoxon signed-rank test because the data does not follow a normal distribution. The results showed that optimizing the CIH algorithm using the Tabu Search algorithm significantly resulted in routes with shorter travel distances than using the CIH algorithm alone. This finding proves that optimizing the CIH algorithm with Tabu Search increases the effectiveness of finding the shortest route.The shortest route finding problem is a significant topic in graph theory and combinato-rial optimization, with wide applications in logistics, transportation, and scheduling. Thisresearch aims to improve the quality of the solution and time efficiency in solving the Trav-eling Salesman Problem (TSP) by optimizing the Cheapest Insertion Heuristic (CIH) algo-rithm using the application of the Tabu Search algorithm. The CIH algorithm constructs aninitial solution by inserting points based on minimum weight. At the same time, the TabuSearch algorithm is applied to enhance the solution by avoiding local optima using a tabulist mechanism. The research data, consisting of the distances between parking retributioncollection points by the Malang City Transportation Agency in Sukun Sub-district, were ob-tained from Google Maps. The algorithm performance evaluation is done by comparing thetotal mileage before and after optimization, and statistically analyzed using the Wilcoxonsigned-rank test because the data does not follow a normal distribution. The results showedthat optimizing the CIH algorithm using the Tabu Search algorithm significantly resulted inroutes with shorter travel distances than using the CIH algorithm alone. This finding provesthat optimizing the CIH algorithm with Tabu Search increases the effectiveness of findingthe shortest route.

Characterizing brain anatomical connections using diffusion weighted MRI and graph theory