Graph Connectivity Research Articles

The Structural Properties of (2, 6)-Fullerenes

A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler’s formula. The (2,6)-fullerene comes from Došlić’s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Došlić showed that the (k,6)-fullerenes only exist for k=2, 3, 4, or 5, and some of the structural properties of (k,6)-fullerene for k=3, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of (2,6)-fullerene. We discover that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable (F is called n-extendable (|V(F)|≥2n+2) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (k≥1) if the deleting of any i (0≤i≤k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if F−S is connected and has no perfect matchings, where F−S denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by ak(F), is the cardinality of a smallest anti−Kekulé set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti−Kekulé number 4. Further we mainly prove that there exists a (2,6)-fullerene F having fF hexagonal faces, where fF is related to the two parameters n and m.

Component Edge Connectivity and Extra Edge Connectivity of Bubble-Sort Star Graphs

Connectivity is an important parameter for evaluating the reliability and stability of an interconnection network. Based on the edge connectivity, more refined connectivities have been proposed. The [Formula: see text]-component edge connectivity [Formula: see text] and the [Formula: see text]-extra edge connectivity [Formula: see text] are two important parameters to assess the robustness of an interconnection network, which received attention extensively. In this paper, we determine the [Formula: see text]-component edge connectivity and the [Formula: see text]-extra edge connectivity of bubble-sort star graphs [Formula: see text]. For [Formula: see text]-component edge connectivity, we prove that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. For [Formula: see text]-extra edge connectivity, we prove that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text].

Outerplane bipartite graphs with isomorphic resonance graphs

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Three additional characterizations are expressed in terms of resonance digraphs, via local structures of inner duals, as well as using distributive lattices on the set of order ideals of posets defined on inner faces of 2-connected outerplane bipartite graphs.

Convex grid drawings of planar graphs with constant edge-vertex resolution

In this work, we continue the study of the area required for convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a vertex and any non-incident edge in the drawing is lower bounded by a constant that does not depend on the size of the graph. We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces, and computes a convex straight-line drawing with edge-vertex resolution at least 12 on an integer grid of size (n−2+a)×(n−2+a), where a=min⁡{n−3,f}. Our result improves the previously best-known area bound of (3n−7)×(3n−7)/2 by Chrobak, Goodrich and Tamassia.

Brain network classification based on dynamic graph attention information bottleneck

Background and ObjectivesGraph neural networks (GNN) have demonstrated remarkable encoding capabilities in the context of brain network classification tasks. They excel at uncovering hidden static connections between brain states. However, brain network signals can be influenced by physiological traits and external variables during clinical detection, resulting in noisy brain graphs. Additionally, many existing algorithms for brain networks primarily focus on static topologies determined by threshold-based criteria, thereby overlooking the real-time variability in brain channel connectivity. These sources of noise and the persistence of static structures inevitably hinder the effective exchange of information during brain network computations. MethodsTo address these challenges, we propose a novel framework called the dynamic graph attention information bottleneck (DGAIB). This framework is designed to dynamically enhance the input raw brain graph structure from the perspective of information theory and graph theory. First, we employ the Spearman function to construct a raw graph. Then, we use a graph information bottleneck (GIB) to optimize the internal graph connections by selectively masking redundant feature embeddings. Finally, we enhance the feature aggregation of each brain state by utilizing a graph attention network (GAT), which promotes improved information exchange among distinct brain regions within the model. These processed representations serve as input for subsequent classification tasks. Experiment and ResultsWe systematically evaluated the robustness and generalizability of our proposed framework through a series of experiments. This evaluation included patient-specific experiments using the electroencephalography (EEG)-based CHB-MIT dataset and cross-patient experiments leveraging the functional magnetic resonance imaging (fMRI)-based ABIDE-I dataset from multiple perspectives.

Connectivity comparison of uniform quantization-based graph transformation and its application in spectrum sensing

In the uniform quantization-based graph transformation of random sequences, the connectivity of the graph is an important feature and can be considered as a basis for designing graph domain processing algorithms. However, the current research on graph transformation focuses mostly on analysing the majorization orders of different input distributions to achieve a comparison of graph connectivity. Generally, calculating majorization order between two candidate probability density functions requires two inversions and one integration for each probability density function, and obtaining an analytical solution is extremely difficult. Even if an analytical solution can be obtained, considering the impact of function transformations such as normalization, analysing the reservation of majorization order is challenging. However, the dispersive order calculation only involves one inversion and differentiation of the distribution functions, and it implies majorization under certain conditions. Consequently, the objectives of this study were to investigate the dispersive order and its order reservation between two random sequences with different probability distributions and to compare the differences in connectivity between the graphs generated under the same conditions, such as the number of uniform quantization levels and sample size. In addition, the properties of complete graphs generated from random sequences were evaluated, including the influence of the probability vector of vertices on the graphs, the sample size and number of vertices on the probability of generating complete graphs. As an application, a graph-based spectrum sensing algorithm based on the block-maxima of the power spectrum was formulated. Extensive simulations showed that the proposed algorithm outperforms the existing graph-based sensing algorithms both in detection performance and the computational complexity.

HGNNLDA: Predicting lncRNA-Drug Sensitivity Associations via a Dual Channel Hypergraph Neural Network.

Drug sensitivity is critical for enabling personalized treatment. Many studies have shown that long non-coding RNAs (lncRNAs) are closely related to drug sensitivity because lncRNAs can regulate genes related to drug sensitivity to affect drug efficacy. Exploring lncRNA-drug sensitivity associations has important implications for drug development and disease treatment. However, identifying lncRNA-drug sensitivity associations based on traditional biological approaches is small-scale and time-consuming. In this work, we develop a dual-channel hypergraph neural network-based method named HGNNLDA to infer unknown lncRNA-drug sensitivity associations. To our best knowledge, HGNNLDA is the first computational framework to predict lncRNA-drug sensitivity associations. HGNNLDA applies the hypergraph neural network to obtain high-order neighbor information on the lncRNA hypergraph and the drug hypergraph, respectively, and utilizes a joint update mechanism to generate lncRNA embeddings and drug embeddings. In traditional graphs, an edge contains only two nodes. However, hyperedges in hypergraphs can contain any number of nodes and hypergraphs can well describe the higher-order connectivity of the lncRNA-drug bipartite graphs. The comprehensive experimental results show that HGNNLDA significantly outperforms the other six state-of-the-art models. Case studies on two drugs further illustrate that HGNNLDA is an effective tool to predict lncRNA-drug sensitivity associations.

Study of temporal changes in the hydrographic network of small mountain rivers in the Ile Alatau, Kazakhstan

Abstract The article presents the outcomes of an assessment of hydrographic network changes within the Almaty city, utilizing geographic information system (GIS) technology and Earth remote sensing data. Two gauge stations were selected along the main rivers within the Almaty city. To identify distinctive alterations in these rivers, hydrological data series encompassing the maximum runoff from 1970 to 2021 were collected and subjected to statistical analysis. Differential integral curves were constructed to pinpoint periods corresponding to peak and trough runoff levels. For each of these identified periods, the processing of satellite imagery allowed for the computation of meandering coefficients for the river channels. Additionally, refinements were made to the slope values of the rivers during these same timeframes, and connectivity graphs were established to elucidate the relationship between slope and liquid runoff for each period. The analysis encompassed an assessment of the impact of anthropogenic factors on both artificial and natural bodies of water while also considering shifts in the boundaries of the Almaty city. The findings derived from this study have practical applications in the planning and design of water supply systems and in the implementation of measures aimed at mitigating the adverse effects of anthropogenic factors on water bodies.

Taming Connectedness in Machine-Learning-Based Topology Optimization with Connectivity Graphs

Despite the remarkable advancements in machine learning (ML) techniques for topology optimization, the predicted solutions often overlook the necessary structural connectivity required to meet the load-carrying demands of the resulting designs. Consequently, these predicted solutions exhibit subpar structural performance because disconnected components are unable to bear loads effectively and significantly compromise the manufacturability of the designs.In this paper, we propose an approach to enhance the topological accuracy of ML-based topology optimization methods by employing a predicted dual connectivity graph. We show that the tangency graph of the Maximal Disjoint Ball Decomposition (MDBD), which accurately captures the topology of the optimal design, can be used in conjunction with a point transformer network to improve the connectivity of the design predicted by Generative Adversarial Networks and Convolutional Neural Networks. Our experiments show that the proposed method can significantly improve the connectivity of the final predicted structures. Specifically, in our experiments the error in the number of disconnected components was reduced by a factor of 4 or more without any loss of accuracy. We demonstrate the flexibility of our approach by presenting examples including various boundary conditions (both seen and unseen), domain resolutions, and initial design domains. Importantly, our method can seamlessly integrate with other existing deep learning-based optimization algorithms, utilize training datasets with models using any valid geometric representations, and naturally extend to three-dimensional applications.

Reconstruction and edge reconstruction of triangle-free graphs

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs {G−v:v∈V(G)}. Let diam(G) and κ(G) denote the diameter and the connectivity of a graph G, respectively, and let G2:={G:diam(G)=2} and G3:={G:diam(G)=diam(G‾)=3}. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in G2∪G3. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph G in G2∪G3 with κ(G)=2. Moreover, they asked whether the result still holds if κ(G)≥3. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in G2∪G3 which contain triangles.) The case when G∈G3 and κ(G)≥3 was recently confirmed by Devi Priya and Monikandan. In this paper, we further show the Reconstruction Conjecture holds for every triangle-free graph G in G2 with κ(G)=3. We also prove similar results about the Edge Reconstruction Conjecture.

The maximum number of complete multipartite subgraphs in graphs with given circumference or matching number

The circumference c(G) of a graph G is the length of a longest cycle in G and the matching number α′(G) is the maximum size of a matching in G. In 1959, Erdős and Gallai determined the maximum size of graphs with given c(G) or α′(G). Let Kr1,…,rs denote the complete multipartite graph with class sizes r1,…,rs and δ(G) be the minimum degree of G. In this paper, we determine the maximum number of copies of Kr1,…,rs in a 2-connected n-vertex graph G with given c(G) and δ(G)≥k. The corresponding problem for an n-vertex graph G with given α′(G) and δ(G)≥k is also addressed. As a corollary of our main results, we determine the maximum number of copies of Kr1,…,rs in an n-vertex graph with given c(G) or α′(G). In addition, the corresponding problems for graphs with given detour order are solved as well.

Non-Centralised Balance Dispatch Strategy in Waked Wind Farms through a Graph Sparsification Partitioning Approach

A novel non-centralised dispatch strategy is presented for wake redirection to optimise large-scale offshore wind farms operation, creating a balanced control between power production and fatigue thrust loads evenly among the wind turbines. This approach is founded on a graph sparsification partitioning strategy that takes into account the impact of wake propagation. More specifically, the breadth-first search algorithm is employed to identify the subgraph based on the connectivity of the wake direction graph, while the PageRank centrality computation algorithm is utilised to determine and rank scores for the shared turbines’ affiliation with the subgraphs. By doing so, the wind farm is divided into smaller subsets of partitioned turbines, resulting in decoupling. The objective function is then formulated by incorporating penalty terms, specifically the standard deviation of fatigue thrust loads, into the maximum power equation. Meanwhile, the non-centralisation sequential quadratic programming optimisation algorithm is subsequently employed within each partition to determine the control actions while considering the objectives of the respective controllers. Finally, the simulation results of case studies prove to reduce computational costs and improve wind farm power production by balancing accumulated fatigue thrust loads over the operational lifetime as much as possible.

On the bond polytope

Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

The cubic vertices of solid minimal bricks

A 3-connected graph is a brick if, for any two vertices u and v, G−u−v has a perfect matching. A brick G is minimal if G−e is not a brick for every edge e of G. Norine and Thomas [J. Combin. Theory Ser. B, 96(4) (2006), pp. 505-513.] conjectured that there exists α>0 such that every minimal brick G contains at least α|V(G)| cubic vertices.A brick is solid if for any two disjoint odd cycles C1 and C2, G−V(C1∪C2) has no perfect matching. In this paper, by proving that every solid minimal brick G has at least 2|V(G)|/5 cubic vertices, we obtain the first non-trivial class of graphs that support the above conjecture.

AB/Push-Pull method for distributed optimization in time-varying directed networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Multiscale graph neural network autoencoders for interpretable scientific machine learning

The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers.

Wheeled Robot Formation Control under Uncertainty Including Adaptive Rejection of Harmonic Disturbances with Arbitrary Frequencies

This work addresses an adaptive regulation problem for formation control of uncertain unicycle-like dynamical robots. The control goal is to drive a finite number of robots to a prescribed formation despite the presence of multi-sinusoidal input disturbances of unknown frequencies and despite uncertainties in the unicycle model, (mass and inertia’s are unknown). Numerical Simulations were done to illustrate the viability of the suggested strategy. The only measurements available for control are the orientation of the unicycles and the relative positions and velocities according to a given connectivity graph.

Deep orientated distance-transform network for geometric-aware centerline detection

The detection of structure centerlines from imaging data plays a crucial role in the understanding, application and further analysis of many diverse problems, such as road mapping, crack detection, medical imaging and biometric identification. In each of these cases, pixel-wise segmentation is not sufficient to understand and quantify overall graph structure and connectivity without further processing that can lead to compound error. We thus require a method for automatic extraction of graph representations of patterning. In this paper, we propose a novel Deep Orientated Distance-transform Network (DODN), which predicts the centerline map and an orientated distance map, comprising orientation and distance in relation to the centerline and allowing exploitation of its geometric properties. This is refined by jointly modeling the relationship between neighboring pixels and connectivity to further enhance the estimated centerline and produce a graph of the structure. The proposed approach is evaluated on a diverse range of problems, including crack detection, road mapping and superficial vein centerline detection from infrared/ color images, improving over the state-of-the-art by 2.1%, 10.9% and 17.3%/ 4.6% respectively in terms of quality, demonstrating its generalizability and performance in a wide range of mapping problems.

Diameter and Connectivity of Finite Simple Graphs

Diameter and Connectivity of Finite Simple Graphs

A polygonal buildings aggregation method considering obstacle elements and visual clarity

A polygonal building aggregation method is proposed in this study to address the shortcomings of existing aggregation methods that mostly focus on maintaining a few features, ignoring the requirement for a clear expression. First, visually conglutinated buildings were clustered considering obstacle elements. Unreasonable triangles, starting from the marginal triangles, were removed to extract the visual conglutinate areas. Bridging modes were designed and local shape features were considered to bridge the visual conglutinate areas. Then, the contours of the aggregated results were established through a minimum loop search on the path connection graph. An experiment was conducted using the data of OpenStreetMap from Nanjing, and the comparative experimental results showed that the proposed method preserved the aggregated results of right-angled and locally shaped features. The advantages of improving the visual clarity of the results were verified by introducing degrees of visual conglutination and complexity.