Problem In Graph Theory Research Articles

Optimization of Urban Distribution Centres: A Multi-Stage Dynamic Location Approach

Customer demand is dynamic and changeable; thus, optimality of the enterprise’s initial location cannot be guaranteed throughout the planning period in order to minimize site selection cost and maximize service reliability in the whole operation cycle. The enterprise planning period is divided into different stages, and a static location model is established at the fixed stage. In addition, a multi-stage dynamic location model is established by introducing the transfer cost between adjacent stages. To reduce the difficulty of solving the dynamic location model, first, we determined the optimal site selection and allocation strategy for each stage. Second, we designed a novel method that transforms the multi-stage dynamic location problem into the shortest path problem in graph theory. Finally, the Dijkstra algorithm was used to find the optimal dynamic location sequence so that its cumulative cost was the lowest in the whole planning period. Through a case study in China, we compare the costs of static and dynamic locations and the location cost under different objectives. The results show that this dynamic location generates more income (as it reduces cost) in comparison to the previous static location, and different location objectives have a substantial influence on location results. At the same time, the findings indicate that exploring the problem of enterprise location from a dynamic perspective could help reduce the operating cost and resources from a sustainable development perspective.

Clique number of Xor products of Kneser graphs

In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in “A general 2-part Erdős-Ko-Rado theorem” by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic KG(N,k) Kneser graphs. Denote this number with f(k,N). We give lower and upper bounds on f(k,N), and we solve the problem up to a constant deviation depending only on k, and find the exact value for f(2,N) if N is large enough. Also we compute that f(k,k2) is asymptotically equivalent to k2.

Asynchronous Event-Based Set Stabilization of Logical Control Networks and its Applications in Finite-Field Networks

In this article, the asynchronous event-triggered control mechanism (AETCM) is introduced to the set stabilization problem of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -valued logical control networks (KVLCNs). Under the proposed AETCM, each node of the system updates its event-triggered instants independently. In this case, how to coordinate the behaviors of various nodes is a challenge for us. As for this article, we first transform the set stabilization problem of KVLCNs into the shortest path problem in graph theory and then construct a weighted directed hypergraph. Next, an improved heap-optimized Dijkstra algorithm is designed for the optimal asynchronous event-triggered controller. Furthermore, we study set stabilization of KVLCNs with time delays. In terms of applications, how to flexibly adapt the AETCM to specific models is a difficult problem. In this article, as applications, the consensus of finite-field networks with arbitrary communication delays and synchronization of KVLCNs is investigated. Finally, two numerical examples are shown to illustrate the effectiveness of the proposed algorithms.

A study of applications of graph colouring in various fields

Graph theory is an important branch of applied mathematics with a lot of applications in many fields. Graph theory has a broad scale of applications in many practical situations. Graph coloring is one decent approach which can deal with many problems of graph theory. The main aim of this paper is to present the importance of graph coloring ideas in various fields for researchers that the concept of graph theory can be used by them. In this paper an overview is presented especially to project the idea of graph theory and applications of graph colouring.

Hamiltonian cycles in annular decomposable Barnette graphs

Barnette’s conjecture is an unsolved problem in graph theory. The problem states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette) graph is Hamiltonian. Partial results have been derived with restrictions on the number of vertices, several properties of face-partitions and dual graphs of Barnette graphs, while some studies focus just on structural characterizations of Barnette graphs. Noting that spider web graphs are a subclass of Annular Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB graphs and their annular-connected subclass (ADB-AC graphs). We show that ADB-AC graphs can be generated from the smallest Barnette graph (B 0) using recursive edge operations. We derive several conditions assuring the existence of Hamiltonian cycles in ADB-AC graphs without imposing restrictions on the number of vertices, face size or any other constraints on the face partitions. We show that there can be two types of annuli in ADB-AC graphs, ring annuli and block annuli. Our main result is, ADB-AC graphs having non-singular sequences of ring annuli are Hamiltonian.

The largest hole in sparse random graphs

AbstractWe show that for any with , with high probability, the size of a largest induced cycle in the random graph is . This settles a long‐standing open problem in random graph theory.

Turán numbers for hypergraph star forests

Fix a graph F. We say that a graph is F-free if it does not contain F as a subhypergraph. The Turán number of F, denoted ex(n,F), is the maximum number of edges possible in an n-vertex F-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [Electron. J. of Combin 20 (2016)] that determines ex(n,F) for F a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in each case we prove an asymptotic result for all reasonable parameters defining our “star forests”.

Eccentric connectivity coindex of bipartite graphs

The eccentric connectivity coindex of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the edge set of [Formula: see text], [Formula: see text] is the eccentricity of [Formula: see text] and [Formula: see text] is the degree of [Formula: see text] in [Formula: see text]. We present lower bounds on the eccentric connectivity coindex for bipartite graphs of given order and vertex connectivity, order and matching number, order and odd diameter. The extremal graphs are presented as well. To the best of our knowledge, our extremal graphs of given order and vertex connectivity are not extremal graphs for any other research problem in graph theory.

Dominating Broadcasts in Fuzzy Graphs

Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph GF:(V,σ,μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ*, a dominating broadcast always exists, where σ* is {v∈V|σ(v)&gt;0}. In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γb(GF)≤min{r(GF),γ(GF)}, where γb(GF) is the broadcast domination number, r(GF) is the radius, and γ(GF) is domination numbers in fuzzy graph GF, with |σ*|&gt;1.

Types of fuzzy graph colouring and polynomial ideal theory

The graph colouring problem (GCP) is one of the most important optimisation problems in graph theory. In real life scenarios, many applications of graph colouring are fuzzy in nature. Fuzzy set and fuzzy graph can manage the uncertainty, associated with the information of a problem, where conventional mathematical models/graph may fail to reveal satisfactory result. To include those fuzzy properties in solving those types of problems, we have extended the various types of classical graph colouring methods to fuzzy graph colouring methods. In this study, we describe three basic types of fuzzy graph colouring methods namely, fuzzy vertex colouring, fuzzy edge colouring and fuzzy total colouring. We introduce a method to colour the vertices of the fuzzy graph using the polynomial ideal theory and find the fuzzy vertex chromatic number of the fuzzy graph. A practical example of scheduling committees meeting is given to demonstrate our proposed algorithm.

Maximum Colored Cuts in Edge‐Colored Complete Graphs

Max‐Cut problem is one of the classical problems in graph theory and has been widely studied in recent years. Maximum colored cut problem is a more general problem, which is to find a bipartition of a given edge‐colored graph maximizing the number of colors in edges going across the bipartition. In this work, we gave some lower bounds on maximum colored cuts in edge‐colored complete graphs containing no rainbow triangles or properly colored 4‐cycles.

Vertex Cover Optimization Using a Novel Graph Decomposition Approach

The minimum vertex cover problem (MVCP) is a well-known combinatorial optimization problem of graph theory. The MVCP is an NP (nondeterministic polynomial) complete problem and it has an exponential growing complexity with respect to the size of a graph. No algorithm exits till date that can exactly solve the problem in a deterministic polynomial time scale. However, several algorithms are proposed that solve the problem approximately in a short polynomial time scale. Such algorithms are useful for large size graphs, for which exact solution of MVCP is impossible with current computational resources. The MVCP has a wide range of applications in the fields like bioinformatics, biochemistry, circuit design, electrical engineering, data aggregation, networking, internet traffic monitoring, pattern recognition, marketing and franchising etc. This work aims to solve the MVCP approximately by a novel graph decomposition approach. The decomposition of the graph yields a subgraph that contains edges shared by triangular edge structures. A subgraph is covered to yield a subgraph that forms one or more Hamiltonian cycles or paths. In order to reduce complexity of the algorithm a new strategy is also proposed. The reduction strategy can be used for any algorithm solving MVCP. Based on the graph decomposition and the reduction strategy, two algorithms are formulated to approximately solve the MVCP. These algorithms are tested using well known standard benchmark graphs. The key feature of the results is a good approximate error ratio and improvement in optimum vertex cover values for few graphs.

Perimeter Degree Technique for the Reduction of Routing Congestion during Placement in Physical Design of VLSI Circuits

When used in conjunction with the current floorplan and the optimization technique in circuit design engineering, this research allows for the evaluation of design parameters that can be used to reduce congestion during integrated circuit fabrication. Testing the multiple alternative consequences of IC design will be extremely beneficial in this situation, as will be demonstrated further below. If the importance of placement and routing congestion concerns is underappreciated, the IC implementation may experience significant nonlinear problems throughout the process as a result of the underappreciation of placement and routing congestion concerns. The use of standard optimization techniques in integrated circuit design is not the most effective strategy when it comes to precisely estimating nonlinear aspects in the design of integrated circuits. To this end, advanced tools such as Xilinx VIVADO and the ICC2 have been developed, in addition to the ICC1 and VIRTUOSO, to explore for computations and recover the actual parameters that are required to design optimal placement and routing for well‐organized and ordered physical design. Furthermore, this work employs the perimeter degree technique (PDT) to measure routing congestion in both horizontal and vertical directions for a silicon chip region and then applies the technique to lower the density of superfluous routing (DSR) (PDT). Recently, a metaheuristic approach to computation has increased in favor, particularly in the last two decades. It is a classic graph theory problem, and it is also a common topic in the field of optimization. However, it does not provide correct information about where and how nodes should be put, despite its popularity. Consequently, in conjunction with the optimized floorplan data, the optimized model created by the Improved Harmonic Search Optimization algorithm undergoes testing and investigation in order to estimate the amount of congestion that occurs during the routing process in VLSI circuit design and to minimize the amount of congestion that occurs.

The spectral radius of graphs with given independence number

As a classic problem of spectral graph theory, Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. In this paper, we focus on this problem for graphs with fixed independence number. We first obtain the maximum spectral radius and unique extremal graph among all bipartite graphs of order n with independence number α. Secondly, we study the minimum spectral radius of graphs with fixed independence number. Let Gn,α be the set of connected graphs of order n with independence number α. We show that tree has the minimum spectral radius over all graphs in Gn,α provided that α≥⌈n2⌉, and determine all the extremal graphs in Gn,n−4.

Divide-and-conquer based all spanning tree generation algorithm of a simple connected graph

All spanning tree generation of a simple connected graph is a well-approached problem in graph theory. In this paper, an algorithm based on a new technique, namely divide-and-conquer, has been proposed. The performance of the proposed algorithm has also been benchmarked against several existing algorithms, including one algorithm implemented in parallel, in this domain. The basis of comparison is on the number of circuits generated and CPU time taken by each of the algorithms compared, using a set of randomly generated graph instances on a common platform.

An Update-Based Maximum Column Distance Coding Scheme for Index Coding

In this paper, we propose a new scalar linear coding scheme for the index coding problem called update-based maximum column distance (UMCD) coding scheme. The central idea in each transmission is to code messages such that one of the receivers with the minimum size of side information is instantaneously eliminated from unsatisfied receivers. One main contribution of the paper is to prove that the other satisfied receivers can be identified after each transmission, using a polynomial-time algorithm solving the well-known maximum cardinality matching problem in graph theory. This leads to determining the total number of transmissions without knowing the coding coefficients. Once this number and what messages to transmit in each round are found, we then propose a method to determine all coding coefficients from a sufficiently large finite field. We provide concrete instances where the proposed UMCD coding scheme has a better broadcast performance compared to the most efficient existing coding schemes, including the recursive scheme (Arbabjolfaei and Kim, 2014) and the interlinked-cycle cover (ICC) scheme (Thapa <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2017). We prove that the proposed UMCD coding scheme performs at least as well as the MDS coding scheme in terms of broadcast rate. By characterizing two classes of index coding instances, we show that the gap between the broadcast rates of the recursive and ICC schemes and the UMCD scheme grows linearly with the number of messages. Then, we extend the UMCD coding scheme to its vector version by applying it as a basic coding block to solve the subinstances.

Martingales and the characteristic functions of absorption time on bipartite graphs.

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald’s martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ1 and λ2 the largest and the second largest eigenvalues of the adjacency matrix A(G) of G, respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on λ12k+λ22k of n-vertex {C3,C5,…,C2k+1}-free graphs is established, where k is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most 2k+1 in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of n-vertex non-bipartite graphs with odd girth at least 2k+3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

On fast enumeration of maximal cliques in large graphs

Maximal Clique Enumeration (MCE) is a fundamental and challenging problem in graph theory and various network applications. Numerous algorithms have been proposed in the past decades, however, only a few of them focus on improving the practical efficiency in large graphs. To this end, we propose an efficient algorithm called FACEN based on the Bron–Kerbosch framework. To optimize the memory and time consumption, we apply a hybrid data structure with adjacency list and partial adjacency matrix, and introduce a dynamic pivot selection rule based on the degeneracy order. FACEN is evaluated on a total of 64 benchmark instances from various sources. Computational results indicate that the proposed algorithm is highly competitive with the current leading MCE methods. In particular, our algorithm is able to enumerate all maximal cliques on the tested real-world social networks with millions of vertices and edges. For very large graphs, we provide an additional experiment for solving the MCE variant with lower bound, and investigate the benefits of FACEN.

Flow Path Resistance in Heterogeneous Porous Media Recast into a Graph-Theory Problem

This work aims to describe the spatial distribution of flow from characteristics of the underlying pore structure in heterogeneous porous media. Thousands of two-dimensional samples of polydispersed granular media are used to (1) obtain the velocity field via direct numerical simulations, and (2) conceptualize the pore network as a graph in each sample. Analysis of the flow field allows us to distinguish preferential from stagnant flow regions and to quantify how channelized the flow is. Then, the graph’s edges are weighted by geometric attributes of their corresponding pores to find the path of minimum resistance of each sample. Overlap between the preferential flow paths and the predicted minimum resistance path determines the accuracy in individual samples. An evolutionary algorithm is employed to determine the “fittest” weighting scheme (here, the channel’s arc length to pore throat ratio) that maximizes accuracy across the entire dataset while minimizing over-parameterization. Finally, the structural similarity of neighboring edges is analyzed to explain the spatial arrangement of preferential flow within the pore network. We find that connected edges within the preferential flow subnetwork are highly similar, while those within the stagnant flow subnetwork are dissimilar. The contrast in similarity between these regions increases with flow channelization, explaining the structural constraints to local flow. The proposed framework may be used for fast characterization of porous media heterogeneity relative to computationally expensive direct numerical simulations.Article HighlightsA quantitative assessment of flow channeling is proposed that distinguishes pore-scale flow fields into preferential and stagnant flow regions.Geometry and topology of the pore network are used to predict the spatial distribution of fast flow paths from structural data alone.Local disorder of pore networks provides structural constraints for flow separation into preferential v stagnant regions and informs on their velocity contrast.