The number of edges in graphs with bounded clique number and circumference

Abstract This paper proposes a novel system to conduct visual servoing of a mobile robot using multiple uncalibrated, wall-mounted cameras. Specifically, we utilise a constellation of such sensors to cover a wide area by allowing robot control to be passed between cameras in regions where their fields of view overlap. This method, in conjunction with the fact that all computing is also executed offboard, allows for simpler and cheaper robots to be deployed in controlled and finite spaces. Our method simplifies the natural installation cycle of a newly deployed camera network, eliminating the need for explicit camera positioning or orientation, both globally (relative to a building plan) and locally (among viewpoints). Our system memorises pixel-wise topological connections between viewpoints by leveraging natural human exploration of the environment. We detect graph edges through simultaneous detections of the same person across different cameras, allowing us to automatically construct an evolving graph that represents overlapping fields of view within the camera network. In combination with a hybrid-A*-based planner, our approach allows efficient planning and control of robots across a wide area by traversing cameras between areas of overlap. We validate our approach through autonomous traversals in a productive office environment, using a network of six cameras, and compare our performance against both human teleoperation and a traditional Simultaneous Localisation and Mapping (SLAM) approach.

A graph is 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let G be a bipartite 1-planar graph with bipartition sets X and Y. A 1-disk O X drawing of G is a 1-planar drawing such that all vertices of X lie on the boundary of O and all vertices of Y and all edges of G locate in the interior of O , where O is a disk on the plane. The concept was first proposed by Huang, Ouyang and Dong when they solved a conjecture about the edge density of bipartite 1-planar graphs. Additionally, they presented a problem of determining the maximum number of edges in a bipartite graph with a 1-disk O X drawing. In this paper, we solve this problem and prove that every bipartite graph G which has a 1-disk O X drawing has at most 2 | V ( G ) | + | X | − 6 edges. Moreover, we demonstrate that this upper bound is tight, in the sense that there are infinitely many graphs for which this bound is attained exactly.

Heuristic techniques such as A* and IDA* with Manhattan distance or linear conflict as heuristics fall short for larger sliding-tile puzzles. This is quite unfortunate as the sliding-tile puzzle is often used as a heuristic search benchmark. We introduce PuzzleGNN, a size-adaptive graph neural network that learns cross-size heuristics for puzzles ranging from 3 3 to 7 7. By encapsulating tiles and adjacencies as graph nodes and edges and embedding the board size into the model, PuzzleGNN functions as a predictor of optimal step counts for puzzles. After being trained on solutions generated by IDA*, PuzzleGNN achieved R2 0.85 within its trained range, performing 10 to 30 times faster in inference compared to A*, and excelling on mid-sized boards with a mean absolute error (MAE) of 0.62 on 6 6. Nonetheless, while generalizing PuzzleGNN to unseen configurations remains challenging, a promising approach is to fuse neural and admissible heuristics as a means to achieving optimal, scalable search in combinatorial domains.

Framework of a global damage identification method based on multi-source signals and spatiotemporal graph neural network (ST-GNN) is proposed for frame structures. A new architecture of the ST-GNN consisting of a temporal feature encoder, a spatial feature encoder and a classifier is constructed. In the ST-GNN, the structure is represented with a graph defined by graph nodes and graph edges with time sequential signals and modal weighted adjacent matrix, and multi-head attention mechanism is used to fuse the multi-source signals of the graph nodes and edges. The modal weighted adjacent matrix with mode shape information is employed to embed deep structural physical behaviors into the ST-GNN, leading the message-passing process of GNN to be more physically meaningful. A new loss function is designed to solve the imbalanced label problem during training of the damage identification model, and evaluation indicators are used to assess the performance of the model. The global damage identification model of a frame structure is created by means of the presented method, and its performances are compared with the models based on the CNN, LSTM, and GRU machine learning algorithms. It is shown that the global damage identification method based on the multisource signals and the newly developed ST-GNN shows superior capabilities in global damage identification, stability, anti-noise performance, and generalization ability.

Abstract An ( r , z ; g )-mixed graph is a graph containing both edges and darts satisfying the regularity property that each vertex of the graph is incident to r edges, z ingoing and z outgoing darts (called total regularity), and being of oriented girth g , i.e., containing an oriented cycle of length g , and no shorter oriented cycles. The problem addressed in this paper is analogous to the Cage Problem and calls for determining the orders of the smallest totally regular ( r , z ; g )-mixed graphs. We derive several upper and lower bounds on the orders of such minimal graphs, study the relations between these extremal graphs and their non-oriented or digraphical counterparts, and focus on properties of totally regular mixed graphs obtained by replacing some of the edges of the incidence graphs of projective and biaffine planes by darts. We also introduce two constructions based on introducing additional edges or darts into induced subgraphs of these incidence graphs.

Partitioning vertices and edges of graphs into connected subgraphs

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such ’master variables’ are usually highly desirable in treatments of ‘large N’ statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with explicit control over the corrections. We use this advantage to compute the first subleading correction to the Park–Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale–Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c\sqrt{k}n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c\sqrt{k}n, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2-- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

Correspondence analysis (CA) and its covariate-based counterpart, canonical correspondence analysis (CCA), are classic yet popular scaling methods in the natural, social, and biomedical sciences to estimate latent gradients that drive the formation of edges in a bipartite graph. However, these methods struggle to identify latent gradients when they exist in sparse graphs where small subsets of nodes are hyperspecialized to each other. This article proposes a new computational method to prevent hyperspecialized nodes from obscuring latent gradient solutions based on a Markov chain interpretation of the CA eigenvalue problem. This approach identifies small subsets of hyperspecialized nodes with greater precision than traditional graph clustering techniques, and outperforms existing regularization techniques at identifying a latent gradient on a real-world political fundraising network of candidates for U.S. federal office, which spans three decades and includes nearly 20,000 candidates for federal office and 3 million of their donors. Supplementary materials for this article are available online.

The precise recognition of human lower limb movements based on wearable sensors is very important for human-computer interaction. However, the existing methods tend to ignore the dynamic spatial information in the process of executing human lower limb movements, leading to challenges such as reduced decoding accuracy and limited robustness. In this paper, we construct skeleton graph data based on inertial measurement unit (IMU) sensors. Also, a two-branch deep learning model, termed TCNN-MGCHN, is proposed to mine meaningful spatial and temporal feature representations from IMU-based skeleton graph data. Firstly, a temporal convolutional module (consisting of a multi-scale convolutional sub-module and an attention sub-module) is developed to extract temporal feature information with highly discriminative power. Secondly, a multi-scale graph convolutional module and a spatial graph edges’ importance weight assignment method based on body partitioning strategy are proposed to obtain intrinsic spatial feature information between different skeleton nodes. Finally, the fused spatio-temporal features are passed into the classification module to obtain the predicted gait movements and sub-phases. Extensive comparison and ablation studies are conducted on our self-constructed human lower limb movement dataset. The results demonstrate that TCNN-MGCHN delivers superior classification performance compared to the mainstream methods. This study can provide a benchmark for IMU-based human lower limb movement recognition and related deep-learning modeling works.

Generalizing and strengthening a classical result of Vizing, Rautenbach proved a linear relationship between the domination number, the maximum degree, the number of vertices, and the number of edges for graphs with no isolated vertices. The sharpest version of this result was established recently by Henning and the first author. In their paper, the question was raised of establishing the sharpest possible linear-Vizing inequality for other domination parameters; finding the minimum so that for every graph G and a domination parameter γ′. In particular, the question was raised of showing that a key portion of the inequality – the linear-Vizing constant – remained bounded when γ′ was distance-2 domination. In this note, we settle this question in the affirmative. The key ingredient arises from a simple new proof of a result of Henning and Lichiardopol bounding the distance-k domination number.

Abstract The effective analysis of high-dimensional Electronic Health Record (EHR) data, with substantial potential for healthcare research, presents notable methodological challenges. Employing predictive modeling guided by a knowledge graph (KG), which enables efficient feature selection, can enhance both statistical efficiency and interpretability. While various methods have emerged for constructing KGs, existing techniques often lack statistical certainty concerning the presence of links between entities, especially in scenarios where the utilization of patient-level EHR data is limited due to privacy concerns. In this paper, we propose the first inferential framework for deriving a sparse KG with statistical guarantee based on a dynamic log-linear topic model. Within this model, the KG embeddings are estimated by performing singular value decomposition on the empirical pointwise mutual information matrix, offering a scalable solution. We then establish entrywise asymptotic normality for the KG low-rank estimator, enabling the recovery of sparse graph edges with controlled type I error. Our work uniquely addresses the under-explored domain of statistical inference about non-linear statistics under the low-rank temporal dependent models, a critical gap in existing research. We validate our approach through extensive simulation studies and then apply the method to real-world EHR data in constructing clinical KGs and generating clinical feature embeddings.

Graph anomaly detection (GAD) refers to identifying abnormal graph nodes or edges that heavily deviate from normal observations. Existing approaches inevitably suffer from the influence of imbalanced data and privacy protection. This shortcoming poses challenges in optimizing node embeddings and detecting multitype anomalies simultaneously, resulting in decreased accuracy of existing GAD models. To address this shortcoming, we introduce a new federated learning model for graph anomaly detection (FedGAD). FedGAD enables collaborative unsupervised learning among decentralized data centers without requiring direct access to the distributed subgraphs. Specifically, FedGAD masks and reconstructs the neighborhood features to enhance the knowledge of node representations. Considering the data diversity across distributed clients, we also design a cross-clients' node representation module that enables nodes to reconstruct neighbors by leveraging information from other clients. Furthermore, we use a multiscale contrastive learning function, which includes both structure-level and contextual-level learning functions, to detect graph anomalies in the condition that subgraphs located at different clients show imbalanced data distributions. Experimental results on seven benchmark datasets demonstrate the superior performance of FedGAD compared with baseline methods, verifying its capability of improving GAD performance.

ProtGeoNet-Pocket is an innovative multimodal prediction framework designed for protein binding site recognition, effectively integrating sequence information, geometric features, and graph-based structural representations. To address the structural complexity of proteins and the diversity of binding pocket shapes, ProtGeoNet-Pocket leverages multiscale structural information for high-precision binding site prediction. It uses a PointNet module to extract geometric features from residue coordinates and enhances them using an attention mechanism. The geometric features are then fused with encoded sequence features and graph edge features. The combined features are fed into a Graph Isomorphism Network (GIN) to capture topological relationships via a message-passing mechanism. ProtGeoNet-Pocket achieved an F1 score of 72.87% on the scPDB training set and demonstrated strong predictive performance across five independent benchmark data sets: COACH420, HOLO4K, SC6K, PDBbind, and ApoHolo. Furthermore, the visualization results confirm a high spatial overlap between the predicted and actual binding sites, demonstrating the superior performance of this method compared to existing ones.

Developing new ethical drugs is exceedingly expensive in terms of both time and resources. A single drug can take up to a decade to bring to market, with costs soaring to over a billion dollars. Drug repositioning has thus become an attractive alternative to the development of new compounds, with growing interest in the use of in silico repositioning predictions. Bipartite graphs and efficient biclique enumeration algorithms can be used to study drug-protein or other pairwise crucial interactions. Extensions of this approach to datasets with three or more divergent data types have been hobbled, however, by a lack of effective analytics. To address this shortcoming, a highly innovative and efficient graph theoretical technique is introduced to impute potential edges (links) in an arbitrary multipartite graph. The utility of this method is demonstrated on five tripartite graphs, each comprised of three partite sets, one each for diseases, drugs, and gene products of interest, and with interpartite edges denoting known interactions or associations. Evidence for the reliability of imputed edges is also reported.

Graph labeling is the assigning of labels represented by integers or symbols to graph elements, edges and/or vertices (or both) of a graph. Consider a simple graph with a vertex-set and an edge-set . The order of graph , denoted by , is the number of vertices on . The prime labeling is a bijective function , such that the labels of any two adjacent vertices in G are relatively prime or , for every two adjacent vertices and in . If a graph can be labeled with prime labeling, then the graph can be said to be a prime graph. A flower graph is a graph formed by helm graph by connecting its pendant vertices (the vertices have degree one) to the central vertex of , such a flower graph is denoted as In this research, we employ constructive and analytical methods to investigate prime labelings on specific graph classes. Definitions, lemmas, and theorems are developed as the main results in this research. The amalgamation is a graph formed by taking all by taking all the and identifying their fixed vertices . If , then we write with . In previous research, it has been shown that the flower graphs , for are prime graphs. Continuing the research, we prove that two classes of amalgamation of flower graphs are prime graphs.

Local efficiency analysis of the emotion regulation network in younger and older adults experiencing sleep deprivation: A task-based fMRI study.

ABSTRACTA ‐star is a complete bipartite graph . A partial ‐star design of order is a pair where is a set of vertices and is a set of edge‐disjoint ‐stars whose vertex sets are subsets of . If each edge of the complete graph with vertex set is in some star in , then is a (complete) ‐star design. We say that is completable if there is a ‐star design such that . In this paper we determine, for all and , the minimum number of stars in an uncompletable partial ‐star design of order .

Abstract The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$ -free graph $G$ is close to the extremal graphs for $H$ , then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$ . As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.