Problem In Graph Theory Research Articles

Combining Genetic Algorithm with Local Search Method in Solving Optimization Problems

This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem—TSP) is considered. The implementations of two approximate algorithms for solving this problem, genetic and memetic, are presented. The main objective of this study is to determine the influence of the local search method versus the influence of the genetic crossover operator on the quality of the solutions generated by the memetic algorithm for the same input data. The results show that when the number of possible Hamiltonian cycles in a graph is increased, the memetic algorithm finds better solutions. The execution time of both algorithms is comparable. Also, the number of solutions that mutated during the execution of the genetic algorithm exceeds 50% of the total number of all solutions generated by the crossover operator. In the memetic algorithm, the number of solutions that mutate does not exceed 10% of the total number of all solutions generated by the crossover operator, summed with those of the local search method.

Cokernel statistics for walk matrices of directed and weighted random graphs

Abstract The walk matrix associated to an $n\times n$ integer matrix $\mathbf{X}$ and an integer vector $b$ is defined by ${\mathbf{W}} \,:\!=\, (b,{\mathbf{X}} b,\ldots, {\mathbf{X}}^{n-1}b)$ . We study limiting laws for the cokernel of $\mathbf{W}$ in the scenario where $\mathbf{X}$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^m$ -torsion part of the cokernel, as a group, when $\mathbf{X}$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ${\mathbb{Z}}[x]$ -module structure. The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

On the hyperspaces of meager and regular continua

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite.In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that | D ( X ) | ∉ { 2,3,4,5,8 , 9 } h t ) for any continuum X. Also, we prove that D(X) is never homeomorphic to ℕ . On the other hand, given a point p ∈ X , we consider the meager composant and the filament composant of p in X, denoted by MXp and FcsX(p), respectively, and we study some relations between MXp and FcsX(p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs ( X ) = { FcsX ( p ) : p ∈ X } can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.

Path Through Specified Nodes and Links in a Network

A constrained shortest route problem in graph theory is about determination of a shortest path between two given nodes of the network that also visits a given set of specified nodes or a set of specified links before arriving to the destination. These earlier approaches did not consider specified elements containing both nodes and links of the given network. This paper finds a shortest path joining the origin node to the destination node, which is constrained to pass through a set of ‘K’ specified elements of the given network, where K1 number of elements represent nodes, 0<K1<K, and K2 number of elements represent links, where K1+K2=K. Alternatively, if the specified elements representing nodes are contained in the set Sn and the remaining elements representing links by the set Sl, and the set of specified elements denoted by the set Se, then Se={Sn ∪ Sl}. Such restricted path will also have real-life applications. Depending upon the configuration of the specified elements, these constrained paths may have loops. The approach discussed in this paper is a heuristic approach, which finds the required constrained path in a real time.

On spectral extrema of graphs with given order and generalized 4-independence number

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph G[S] dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where ψ⩾⌈3n/4⌉. Finally, we identify all the connected n-vertex graphs with generalized 4-independence number ψ∈{3,⌈3n/4⌉,n−1,n−2} having the minimum spectral radius.

Complete solution to open problems on exponential augmented Zagreb index of chemical trees

One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G asEAZ(G)=∑vivj∈E(G)e(didjdi+dj−2)3, where di signifies the degree of vertex vi, and E(G) is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.

Simulating Chemistry on Bosonic Quantum Devices.

Bosonic quantum devices offer a novel approach to realize quantum computations, where the quantum two-level system (qubit) is replaced with the quantum (an)harmonic oscillator (qumode) as the fundamental building block of the quantum simulator. The simulation of chemical structure and dynamics can then be achieved by representing or mapping the system Hamiltonians in terms of bosonic operators. In this Perspective, we review recent progress and future potential of using bosonic quantum devices for addressing a wide range of challenging chemical problems, including the calculation of molecular vibronic spectra, the simulation of gas-phase and solution-phase adiabatic and nonadiabatic chemical dynamics, the efficient solution of molecular graph theory problems, and the calculations of electronic structure.

On Some Distance Spectral Characteristics of Trees

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with “few eigenvalues” is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is “highly” non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions.

Near-term quantum algorithm for solving the MaxCut problem with fewer quantum resources

MaxCut is an NP-hard combinatorial optimization problem in graph theory. The quantum approximate optimization algorithms (QAOAs) offer new methods for solving such problems, which are potentially better than classical schemes. However, the requirement of N qubits for solving graphs with N-vertex in QAOA, coupled with the large-N trainability issue due to barren plateaus, poses a substantial challenge for noisy intermediate-scale quantum (NISQ) devices. Recently, a hybrid quantum–classical algorithm has been proposed for solving semidefinite programs (SDPs), named NISQ SDP Solver (NSS). In this paper, we study the performance of the NSS for solving MaxCut problem via the introduction of a near-term quantum algorithm and execution of experimental simulations. Since the MaxCut problem admits a relaxation into an SDP formulation, the SDP can be solved using NSS, with a subsequent classical post-processing step converting the hybrid density matrix into a MaxCut solution. After performing these steps, the near-term quantum algorithm will also inherit the advantages of NSS. We analyze the algorithm as compared to QAOAs and find that the depth of the quantum circuit is independent of the number of edges on the graph. Our algorithm requires logN qubits and has O(1) circuit depth. This implies that it uses exponentially fewer qubits compared to QAOAs while also requiring a significantly reduced circuit depth to solve the MaxCut problem. To demonstrate the effectiveness of the NSS, we focused on 3-regular graphs, 9-regular graphs, and ER graphs. Numerical results indicate that the quantum algorithm achieves a comparable approximation ratio with classical methods by solving the original high dimensional problem within a lower dimensional ansatz space. Analysis under various initial states shows that the algorithm exhibits a remarkably stable performance.

A characterization for fuzzy strong cut vertices and fuzzy strong cut edges

Cut vertices and cut edges are valuable for analyzing connectivity problems in classical graph theory. However, they cannot deal with certain fuzzy problems. In order to solve this problem, this paper introduces the definitions of fuzzy strong cut vertices and fuzzy strong cut edges, and characterizes fuzzy strong cut vertices and fuzzy strong cut edges in fuzzy trees, complete fuzzy graphs, and fuzzy cycles. Finally, practical applications verify the effectiveness of the theory in network stability analysis.

Optimizing Collaborative Crowdsensing: A Graph Theoretical Approach to Team Recruitment and Fair Incentive Distribution.

Collaborative crowdsensing is a team collaboration model that harnesses the intelligence of a large network of participants, primarily applied in areas such as intelligent computing, federated learning, and blockchain. Unlike traditional crowdsensing, user recruitment in collaborative crowdsensing not only considers the individual capabilities of users but also emphasizes their collaborative abilities. In this context, this paper takes a unique approach by modeling user interactions as a graph, transforming the recruitment challenge into a graph theory problem. The methodology employs an enhanced Prim algorithm to identify optimal team members by finding the maximum spanning tree within the user interaction graph. After the recruitment, the collaborative crowdsensing explored in this paper presents a challenge of unfair incentives due to users engaging in free-riding behavior. To address these challenges, the paper introduces the MR-SVIM mechanism. Initially, the process begins with a Gaussian mixture model predicting the quality of users' tasks, combined with historical reputation values to calculate their direct reputation. Subsequently, to assess users' significance within the team, aggregation functions and the improved PageRank algorithm are employed for local and global influence evaluation, respectively. Indirect reputation is determined based on users' importance and similarity with interacting peers. Considering the comprehensive reputation value derived from the combined assessment of direct and indirect reputations, and integrating the collaborative capabilities among users, we have formulated a feature function for contribution. This function is applied within an enhanced Shapley value method to assess the relative contributions of each user, achieving a more equitable distribution of earnings. Finally, experiments conducted on real datasets validate the fairness of this mechanism.

The homomorphism and coloring of the direct product of signed graphs

Hedetniemi's conjecture, a significant problem in graph theory, was disproved by Shitov in 2019. This breakthrough stimulates us to consider the signed graphs version of Hedetniemi's conjecture, where signed graphs are graphs with an assignment of {±1} to the edges. In this paper, we provide counterexamples to the signed analog of Hedetniemi's conjecture. Nevertheless, we show that the chromatic number of the product is determined by its factors in some instances, i.e., if Γ→Θ and (Θ,π) is balanced, then χ((Γ,σ)×(Θ,π))=χ((Γ,σ)). As a byproduct, we investigate the homomorphisms from a factor of (Γ,σ)×(Θ,π) to the other, say from (Γ,σ) to (Θ,π). We find that if such homomorphisms exist, then the chromatic number of (Γ,+) provides a lower bound for χ((Γ,σ)×(Θ,π)). Moreover, if Γ is a core, then the number of balanced subgraphs isomorphic to (Γ,+) in (Γ,σ)×(Θ,π) is equal to the number of such homomorphisms.

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

Phasor measurement units (PMUs) are deployed at power grid nodes around the transmission grid, determining precise power system monitoring conditions. In real life, it is not realistic to place a PMU at every power grid node; thus, the lowest PMU number is optimally selected for the full observation of the entire network. In this study, the PMU placement model is reconsidered, taking into account single- and multi-capacity placement models rather than the well-studied PMU placement model with an unrestricted number of channels. A restricted number of channels per monitoring device is used, instead of supposing that a PMU is able to observe all incident buses through the transmission connectivity lines. The optimization models are declared closely to the power dominating set and minimum edge cover problem in graph theory. These discrete optimization problems are directly related with the minimum set covering problem. Initially, the allocation model is declared as a constrained mixed-integer linear program implemented by mathematical and stochastic algorithms. Then, the 0/1 integer linear problem is reformulated into a non-convex constraint program to find optimality. The mathematical models are solved either in binary form or in the continuous domain using specialized optimization libraries, and are all implemented in YALMIP software in conjunction with MATLAB. Mixed-integer linear solvers, nonlinear programming solvers, and heuristic algorithms are utilized in the aforementioned software packages to locate the global solution for each instance solved in this application, which considers the transformation of the existing power grids to smart grids.

Signed graphs with exactly two distinct main eigenvalues

An eigenvalue λ of a signed graph S of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. Characterizing signed graphs with exactly k(1≤k≤n) main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Z. Stanić has noticed that a signed graph has exactly one main eigenvalue if and only if it is net-regular, and in this paper, we study signed graphs with exactly two distinct main eigenvalues by studying (0,1,2)-multi-graphs with exactly two distinct main eigenvalues. We partially characterize the case when the basic graphs of the (0,1,2)-multi-graphs are trees, and propose some problems for further research.

Spectral methods on the hydrogen atom

This study investigates the computational implementation of spectral theory and its applications in analyzing complex mathematical problems. It explores the use of modern programming languages and scientific libraries for implementing and visualizing complex mathematical concepts, particularly focusing on spectral theory within quantum physics. The research employs computational methods to address the challenges in interpreting spectral properties, utilizing the Lanczos spectral method for eigenvalue calculation in large, sparse matrices. The results illustrate the effectiveness of these computational techniques in visualizing quantum states, demonstrating the potential of advanced programming in understanding and solving intricate problems in quantum physics and spectral graph theory. The study's findings are significant in bridging computational methods with theoretical spectral analysis, offering a new perspective on the application of computational techniques in scientific research.

Haemers’ Conjecture: An Algorithmic Perspective

Characterizing graphs by their spectra is a fundamental and challenging problem in spectral graph theory, which has received considerable attention in recent years. A major unsolved conjecture in this area is Haemers’ conjecture which states that almost all graphs are determined by their spectra. Despite many efforts, little is known about this conjecture so far. In this paper, we shall consider Haemers’ conjecture from an algorithmic perspective. Based on some recent developments in the generalized spectral characterizations of graphs, we propose an algorithm to find all possible generalized cospectral mates for a given n-vertex graph G, assuming that G is controllable or almost controllable. The experimental results indicate that the proposed algorithm runs surprisingly fast for most graphs with several dozen vertices. Moreover, we observe in the experiment that most graphs are determined by their generalized spectra, e.g., at least 9945 graphs are determined by their generalized spectra among all randomly generated 10,000 graphs on 50 vertices in one experiment. These experimental results give strong evidence for Haemers’ conjecture.

Binding Number, $k$-Factor and Spectral Radius of Graphs

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_{G}(X)\neq V(G)$. The association between the binding number and toughness is intricately interconnected, as both metrics function as pivotal indicators for quantifying the vulnerability of a graph. The Brouwer-Gu Theorem asserts that for any $d$-regular connected graph $G$, the toughness $t(G)$ always at least $\frac{d}{\lambda}-1$, where $\lambda$ denotes the second largest absolute eigenvalue of the adjacency matrix. Inspired by the work of Brouwer and Gu, in this paper, we investigate $b(G)$ from spectral perspectives, and provide tight sufficient conditions in terms of the spectral radius of a graph $G$ to guarantee $b(G)\geq r$. The study of the existence of $k$-factors in graphs is a classic problem in graph theory. Katerinis and Woodall state that every graph with order $n\geq 4k-6$ satisfying $b(G)\geq 2$ contains a $k$-factor where $k\geq 2$. This leaves the following question: which $1$-binding graphs have a $k$-factor? In this paper, we also provide the spectral radius conditions of $1$-binding graphs to contain a perfect matching and a $2$-factor, respectively.

An optimization algorithm for maximum quasi-clique problem based on information feedback model.

The maximum clique problem in graph theory is a well-known challenge that involves identifying the complete subgraph with the highest number of nodes in a given graph, which is a problem that is hard for nondeterministic polynomial time (NP-hard problem). While finding the exact application of the maximum clique problem in the real world is difficult, the relaxed clique model quasi-clique has emerged and is widely applied in fields such as bioinformatics and social network analysis. This study focuses on the maximum quasi-clique problem and introduces two algorithms, NF1 and NR1. These algorithms make use of previous iteration information through an information feedback model, calculate the information feedback score using fitness weighting, and update individuals in the current iteration based on the benchmark algorithm and selected previous individuals. The experimental results from a significant number of composite and real-world graphs indicate that both algorithms outperform the original benchmark algorithm in dense instances, while also achieving comparable results in sparse instances.

A novel approach for calculating single-source shortest paths of weighted digraphs based on rough sets theory.

Calculating single-source shortest paths (SSSPs) rapidly and precisely from weighted digraphs is a crucial problem in graph theory. As a mathematical model of processing uncertain tasks, rough sets theory (RST) has been proven to possess the ability of investigating graph theory problems. Recently, some efficient RST approaches for discovering different subgraphs (e.g. strongly connected components) have been presented. This work was devoted to discovering SSSPs of weighted digraphs by aid of RST. First, SSSPs problem was probed by RST, which aimed at supporting the fundamental theory for taking RST approach to calculate SSSPs from weighted digraphs. Second, a heuristic search strategy was designed. The weights of edges can be served as heuristic information to optimize the search way of $ k $-step $ R $-related set, which is an RST operator. By using heuristic search strategy, some invalid searches can be avoided, thereby the efficiency of discovering SSSPs was promoted. Finally, the W3SP@R algorithm based on RST was presented to calculate SSSPs of weighted digraphs. Related experiments were implemented to verify the W3SP@R algorithm. The result exhibited that W3SP@R can precisely calculate SSSPs with competitive efficiency.

Huawei Cloud Adopts Operations Research for Live Streaming Services to Save Network Bandwidth Cost: The GSCO System

The rapid evolution of cloud computing technologies has instigated a paradigm shift across various traditional industries, with the live streaming sector standing as a compelling exemplification of this transformation. Huawei Cloud, which has become an influential player in the business-to-business live streaming arena, with its services spanning over 60 countries since 2020, is at the forefront of this shift. Amid the flourishing live streaming market, Huawei Cloud faces the dual challenge of satisfying the escalating demand, while managing the mounting operational costs, predominantly associated with the network bandwidth. To offer premium services while minimizing the bandwidth cost, we developed a dynamic traffic allocation system called GSCO. This system was engineered using an array of operations research methodologies such as continuous optimization, integer programming, graph theory, scheduling, and network-flow problem solving, along with state-of-the-art machine learning algorithms. The GSCO system has been proven highly effective in cost optimization, reducing network bandwidth expenses by about 30% and leading to savings exceeding $49.6 million from Q1 2020 to Q3 2022. In addition, it has significantly bolstered Huawei Cloud’s market share, amplifying peak bandwidth from an initial 1.5 terabits per second (Tbps) to a substantial 16 Tbps.