Graph products for configuration processing of space structures

: There are four prominent product graphs in graph theory: Cartesian, strong, direct and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted G o H, for some digraphs G and H. The paper concludes by proving a cancellation property for the lexicographic product of digraphs G, H, A, and B: if G o H = A o B and /V(G)/ = /V(A)/, then G = A. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

Considering a set of intervals on the real line, an interval graph records these intervals as nodes and their intersections as edges. Identifying (i.e. merging) pairs of nodes in an interval graph results in a multiple-interval graph. Given only the nodes and the edges of the multiple-interval graph without knowing the underlying intervals, we are interested in the following questions. Can one determine how many intervals correspond to each node? Can one compute a walk over the multiple-interval graph nodes that reflects the ordering of the original intervals? These questions are closely related to linked-read DNA sequencing, where barcodes are assigned to long molecules whose intersection graph forms an interval graph. Each barcode may correspond to multiple molecules, which complicates downstream analysis, and corresponds to the identification of nodes of the corresponding interval graph. Resolving the above graph-theoretic problems would facilitate analyses of linked-reads sequencing data, through enabling the conceptual separation of barcodes into molecules and providing, through the molecules order, a skeleton for accurately assembling the genome. Here, we propose a framework that takes as input an arbitrary intersection graph (such as an overlap graph of barcodes) and constructs a heuristic approximation of the ordering of the original intervals.

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

Contributors

Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the complement of the intersection graph of subgroups of G.

A graph G is an odd-circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere-zero 3-flow unless G is an odd-circuit tree and H has a bridge. This theorem is a partial result to the Tutte's 3-flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere-zero 3-flow. A byproduct of this theorem is that every bridgeless Cayley graph G = Cay(Γ,S) on an abelian group Γ with a minimal generating set S admits a nowhere-zero 3-flow except for odd prisms. © 2005 Wiley Periodicals, Inc. J Graph Theory

There are numerous applications of graph theory and algebraic graph theory in combinatorial optimization and optimal structural analysis. In this paper, a new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented. Furthermore, the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied. This block diagonalization leads to an efficient method for the eigensolution of adjacency and Laplacian matrices of special graphs. The eigenvalues and eigenvectors are used for efficient nodal ordering and partitioning of large structural models. The present method is far more simple than any existing general approach.

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 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.

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry.

A class of highly symmetric graphs, symmetric cylindrical constructions and their spectra

Social networks are often analyzed through a graph model of the network. The dot product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector av represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if au · av ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph..We consider two measures for diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is measured through the size of the largest independent set of the graph, and clustering is measured through the size of the largest clique. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-complete for d ≥ 3. We show that the clustering problem is polynomial-time solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs.We also consider the situation when two individuals are connected if their preferences are not opposite. This leads to a variant of the standard dot product graph model by taking the threshold t to be zero. We prove in this case that the diversity problem is polynomial-time solvable for any fixed d.KeywordsSocial NetworkMaximum CliqueStructural ResultProduct GraphInterval GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.

The fractional versions of graph-theoretic invariants multiply the range of applications in scheduling, assignment and operational research problems. For this interesting aspect of fractional graph theory, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractional local metric dimension of $G$. We study the properties and bounds of fractional local metric dimension of graphs and give some characterization results. We determine the fractional local metric dimension of strong and Cartesian product of graphs.

Erratum to “The thickness of amalgamations and Cartesian product of graphs" [Discuss. Math. Graph Theory 37(2017) 561–572