Cycles In Graphs Research Articles

Newly defined fuzzy Misbalance Prodeg Index with application in multi-criteria decision-making

<abstract><p>In crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of $ 1 $ or $ 0 $, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multi-criteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices.</p></abstract>

Fair Detour Domination in the Corona of Two Graphs: Optimizing CCTV Camera Installation for Efficient Surveillance

The objective of our research is to analyze the properties of FDD sets in the corona of two graphs and explore their practical application in CCTV camera installation. A set F ⊆ V(G) is considered to be a Fair Detour Dominating set (FDD-set) if it is detour dominating and the number of neighbors within set F is the same for any pair of vertices outside of F. Among these FDD sets, the fγd-set refers to the FDD-set with the smallest number of vertices, and its order defines the Fair Detour Domination number (fγd (G)). We have established that for any arbitrary graphs G1 and G2, fγd (G1 o G2) = |V(G1) iff fd(G2) = 1 we have determined the fγd number of corona products of any connected graph G with the path graph as well as the cycle graph. We also characterized FDD sets in the corona product of two connected graphs and provided a thorough description of how FDD sets can be used in optimizing CCTV camera installation for efficient surveillance.

Navigating the Maize: Cyclic and conditional computational graphs for molecular simulation

Many computational chemistry and molecular simulation workflows can be expressed as graphs. This abstraction is useful to modularize and potentially reuse existing components, as well as provide parallelization and ease...

Results on Grundy Chromatic Number of Join Graph of Graphs

A Grundy k -coloring of a graph G is a proper k -coloring of vertices in G using colors { 1 , 2 , ⋯ , k } such that for any two colors x and y , x < y , any vertex colored y is adjacent to some vertex colored x . The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G , denoted by Γ ( G ) , is the largest integer k , such that there exists a Grundy k -coloring for G . It can be easily seen that Γ ( G ) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G . In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

Construction Of an Ideal Topological Space From Undirected Graphs

Graph theory, which is used effectively in many field from science to liberal arts, has very important place in our lives. As a result of this , the ideal topological structure of the graphs is studied by many researchers. In this paper ,investigate the topological spaces generated by cycle graph. The ideal topological space is obtained by deleting one of the edges of the cycle graph, then we take the largest tree graph through its vertices , we get the ideal topological space.

Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

In Thermodynamics and Statistical Physics, a system’s property is extensive when it grows with the system size. When it happens, the system can be decomposed into separate components, which has been done in many systems with weakly interacting components, such as for various gas models. Similarly, Ruelle conjectured 40 years ago that the Lyapunov exponents (LEs) of some sufficiently large chaotic systems are extensive, which led to study the extensivity properties of chaotic systems with strong interactions. Because of the complexities in these systems, most results achieved so far are restricted to numerical simulations. Here, we derive closed-form expressions for the LEs and entropy rate of coupled maps in finite- and infinite-sized regular graphs, according to the coupling strength, map’s chaoticity, and graph’s spectral properties. We show that this type of system has either 4 or 5 cases for the LEs, depending on the graph’s extreme Laplacian eigenvalues. These cases represent qualitatively different collective behaviours emerging in parameter space, including chaotic synchronisation (N−1 negative LEs) and incoherent chaos (N positive LEs). From the entropy rate, we show that the ring and complete graphs (nearest-neighbour and all-to-all couplings, respectively) are extensive in all parameter regions outside the chaotic synchronisation region. Although our derivations are restricted to one-dimensional maps with constant positive derivative (i.e., chaotic), our approach can be used to find LE and entropy rates for other regular graphs (such as for cyclic graphs) or be the basis for tackling small world graphs via perturbative methods.

Partial and Simultaneous Transitive Orientations via Modular Decompositions

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.

Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph

In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is normal if every edge of G is either rich or poor. In this paper we consider snarks G˜ obtained by a simple superposition of edges and vertices of a cycle C in a snark G. For an even cycle C we show that a normal coloring of G can be extended to a normal coloring of G˜ without changing colors of edges outside C in G. An interesting remark is that this is in general impossible for odd cycles, since the normal coloring of a Petersen graph P10 cannot be extended to a superposition of P10 on a 5-cycle without changing colors outside the 5-cycle. On the other hand, as our colorings of the superpositioned snarks introduce 18 or more poor edges, we are inclined to believe that every bridgeless cubic graph distinct from P10 has a normal coloring with at least one poor edge and possibly with at least 6 if we also exclude the Petersen graph with one vertex being truncated.

Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n, where one edge has weight α∈C, with Re(α)<0, and all the others have weights 1. This paper is a sequel of a previous one where we considered Re(α)∈[0,1] (Grudsky et al., 2022 [12]). We prove that for Re(α)<0 and n>Re(α−1)/Re(α), one eigenvalue is negative while the others belong to [0,4] and are distributed as the function x↦4sin2⁡(x/2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4Re(α)2/(2Re(α)−1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [0,4] and the outlier. We also compute the eigenvectors and their norms.

5‐Coloring reconfiguration of planar graphs with no short odd cycles

AbstractThe coloring reconfiguration graph has as its vertex set all the proper ‐colorings of , and two vertices in are adjacent if their corresponding ‐colorings differ on a single vertex. Cereceda conjectured that if an ‐vertex graph is ‐degenerate and , then the diameter of is . Bousquet and Heinrich proved that if is planar and bipartite, then the diameter of is . (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when is planar and has no 3‐cycles. As a partial solution to this problem, we show that the diameter of is for every planar graph with no 3‐cycles and no 5‐cycles.

THE EARTH’S MAGNETIC FIELD AND THE LARGE-SCALE MAGNETIC FIELD OF THE SUN: THE SOLAR-TERRESTRIAL CONNECTION

The paper presents the results of a joint analysis of the Earth's main magnetic field (core field) BIGRF and the large-scale magnetic field (LSMF) of the Sun. Selected 11- and 22-year periods of LSMF and BIGRF variations are well manifested in both fields and are usually modulated by solar activity. Even 11-year cycles for which the direction of the Sun's magnetic field coincides with the direction of the Earth's magnetic field are characterized by the minimum values of sanspot numbers, and odd cycles with opposite directions of magnetic fields have larger values of sunspot numbers. The rotation rate of two- and four-sector structure of the Sun source of LSMF varied with about 11-year and 22-year cycle. Longer changes in the magnetic fields of the Sun and the Earth with a period of about 75 years have also been revealed. The rotation periods of the Sun source global field (28.0-28.5 days) were maximum at the middle of the 20th century in the period 1940÷1960 years. This maximum of solar activity corresponds to temporal gradient of geomagnetic field. It is shown that the gradient of the geomagnetic field BIGRF depends on the rate of change in the length of the day. So, according to the results of the study, the rotation modes of the Sun and the Earth cause different periodic changes in their magnetic fields.

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive non-asymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. The proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

On the Difference of Atom-Bond Sum-Connectivity and Atom-Bond-Connectivity Indices

The atom-bond connectivity (ABC) index is one of the wellinvestigated degree-based topological indices. The atom-bond sumconnectivity (ABS) index is a modified version of the ABC index, which was introduced recently. The primary goal of the present paper is to investigate the difference between the aforementioned two indices, namely ABS − ABC. It is shown that the difference ABS − ABC is positive for all graphs of minimum degree at least 2 as well as for all line graphs of those graphs of order at least 5 that are different from the path and cycle graphs. By means of computer search, the difference ABS − ABC is also calculated for all trees of order at most 15.

Attribute-Guided Collaborative Learning for Partial Person Re-Identification.

Partial person re-identification (ReID) aims to solve the problem of image spatial misalignment due to occlusions or out-of-views. Despite significant progress through the introduction of additional information, such as human pose landmarks, mask maps, and spatial information, partial person ReID remains challenging due to noisy keypoints and impressionable pedestrian representations. To address these issues, we propose a unified attribute-guided collaborative learning scheme for partial person ReID. Specifically, we introduce an adaptive threshold-guided masked graph convolutional network that can dynamically remove untrustworthy edges to suppress the diffusion of noisy keypoints. Furthermore, we incorporate human attributes and devise a cyclic heterogeneous graph convolutional network to effectively fuse cross-modal pedestrian information through intra- and inter-graph interaction, resulting in robust pedestrian representations. Finally, to enhance keypoint representation learning, we design a novel part-based similarity constraint based on the axisymmetric characteristic of the human body. Extensive experiments on multiple public datasets have shown that our model achieves superior performance compared to other state-of-the-art baselines.

The automorphism groups of some token graphs

The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6≠ 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs.

Computing minimal solutions to the ring loading problem

Given a cycle graph of n nodes and a non-negative demand di,j for each pair of nodes i,j, the ring loading problem with demand splitting (RLPW) asks to determine a routing of these demands such that the maximum load on any edge is minimal. In this work, we present an algorithm for computing minimal solutions to RLPW in O(n2) time. This algorithm, when employed as a subroutine, also improves the runtimes for several other ring loading algorithms by a factor of up to O(n2).

Algebraic Properties of Powers of Edge Ideals of Vertex-Weighted Oriented Cycles

Let [Formula: see text] be any positive integer and [Formula: see text] the edge ideal of a vertex-weighted oriented [Formula: see text]-cycle graph [Formula: see text]. We provide explicit formulas for the regularity and depth of [Formula: see text]. In particular, we find that the regularity of [Formula: see text] is a linear function; [Formula: see text], the set of associated prime ideals of [Formula: see text], equals [Formula: see text]; and the depth of [Formula: see text] is a constant for all [Formula: see text]. We also give some examples to show that these results are related to the direction selection of edges and the weight of vertices.

Graph-Based Generalization of Galam Model: Convergence Time and Influential Nodes

We study a graph-based generalization of the Galam opinion formation model. Consider a simple connected graph which represents a social network. Each node in the graph is colored either blue or white, which indicates a positive or negative opinion on a new product or a topic. In each discrete-time round, all nodes are assigned randomly to groups of different sizes, where the node(s) in each group form a clique in the underlying graph. All the nodes simultaneously update their color to the majority color in their group. If there is a tie, each node in the group chooses one of the two colors uniformly at random. Investigating the convergence time of the model, our experiments show that the convergence time is a logarithm function of the number of nodes for a complete graph and a quadratic function for a cycle graph. We also study the various strategies for selecting a set of seed nodes to maximize the final cascade of one of the two colors, motivated by viral marketing. We consider the algorithms where the seed nodes are selected based on the graph structure (nodes’ centrality measures such as degree, betweenness, and closeness) and the individual’s characteristics (activeness and stubbornness). We provide a comparison of such strategies by conducting experiments on different real-world and synthetic networks.

Towards the Chen-Raspaud conjecture

For an integer k, a homomorphism from a graph G to the Kneser graph K(2k+1,k) is equivalent to assigning to each vertex of G a k-subset of {1,…,2k+1} in a way that adjacent vertices receive disjoint subsets.Chen and Raspaud (2010) [5] conjectured that for every k≥2, every graph G with maximum average degree less than 2k+1k and no odd cycles with fewer than 2k+1 vertices admits a homomorphism to K(2k+1,k). They also showed that the statement is true for k=2. In this note we confirm the conjecture for k=3.

Boundedness and liveness enforcement for labeled Petri nets using transition priority

This paper deals with the supervisory control problem of discrete event systems modeled by labeled Petri nets. The system is originally unbounded. First, the solvability of the problem is confirmed. A necessary condition is given and proven for the existence of a feasible priority-based controller based on the notions of liveness and transition invariants. Next, a cyclic behavior graph is constructed, which shows the reachable markings that guarantee the maximum liveness of the system within a given bound vector. Finally, an on-line control strategy is proposed to enforce boundedness and liveness to the given system by appending priority relations to transitions. The dynamic priority relation changes flexibly according to the current state of the system and enforces the system evolving in a bounded and live manner. In addition, numerical examples are studied to verify the validity of the proposed approach that remains the structure of the plant net and is efficient for on-line control.