Cycles In Graphs Research Articles

Extended commonality of paths and cycles via Schur convexity

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph Kn is asymptotically minimised by the random colouring, or equivalently, tH(W)+tH(1−W)≥21−e(H) holds for every graphon W:[0,1]2→[0,1], where tH(.) denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, tH(W)+tH(1−W)≥tK2(W)e(H)+tK2(1−W)e(H) whenever H is a path or a cycle and W:[0,1]2→R is a bounded symmetric measurable function.This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons W and odd cycles H. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.

Synchronization of Turing patterns in complex networks of reaction–diffusion systems set in distinct domains

We present an innovative complex network of reaction–diffusion systems set in distinct domains, with boundary couplings. The complex network models the evolution of interacting populations living in a heterogeneous and fragmented habitat, whose biological individuals migrate from one patch to another. In our model, the displacements of individuals are described by mixed boundary couplings, involving both the Neumann and Robin boundary conditions, which improve the modeling of migrations by point-wise couplings. We investigate the cases of diffusion in isotropic and anisotropic habitats and establish original sufficient conditions of synchronization in this complex network model, for complete graphs, cyclic graphs and rings of nearest neighbors. In parallel, we apply our theoretical framework to a nonlinear predator–prey model with Leslie–Gower-type functional response and explore numerically the emergence of synchronization on heterogeneous Turing patterns.

The Complex(ity) Landscape of Checking Infinite Descent

Cyclic proof systems, in which induction is managed implicitly, are a promising approach to automatic verification. The soundness of cyclic proof graphs is ensured by checking them against a trace-based Infinite Descent property. Although the problem of checking Infinite Descent is known to be PSPACE-complete, this leaves much room for variation in practice. Indeed, a number of different approaches are employed across the various cyclic proof systems described in the literature. In this paper, we study criteria for Infinite Descent in an abstract, logic-independent setting. We look at criteria based on Büchi automata encodings and relational abstractions, and determine their parameterized time complexities in terms of natural dimensions of cyclic proofs: the numbers of vertices of the proof-tree graphs, and the vertex width—an upper bound on the number of components (e.g., formulas) of a sequent that can be simultaneously tracked for descent. We identify novel algorithms that improve upon the parameterised complexity of the existing algorithms. We implement the studied criteria and compare their performance on various benchmarks.

A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

AbstractErdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half‐integral packing. We prove a far‐reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half‐integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least , cycles of length modulo , cycles intersecting a prescribed set of vertices at least times and cycles contained in given ‐homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

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).