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Wastewater treatment plant (WWTP) discharges of microcontaminants negatively impact freshwater streams, underscoring the need for accurately mapping wastewater-derived contamination in water bodies across Europe (EU). In this study, we present a fast and open-source code for a microcontaminants fate and transport (MFT) model, capable of simulating contamination at a high resolution (15 arc second scale) across the EU. The model was developed using the best publicly available hydrological (HydroSHEDS) and sanitation (UWWTD) datasets and was rigorously calibrated, with a goodness of fit of 77.5% as measured by the R 2. Importantly, the model demonstrated the ability to predict wastewater-derived contamination in water bodies, making it a valuable tool for planning the upgrade of WWTPs and improving the ecological condition of freshwater streams in the EU. • A model is developed to simulate wastewater-based contamination in rivers. • Large-scale free hydrologic and sanitation layers are used. • The model is calibrated obtaining an R2 of 77.5%. • The model is open-source and has fast runtimes. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph and A (G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A (G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3] , the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: "Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?" Suppose that G is singular and the characteristic polynomial of G is x n − k (x k + a 1 x k − 1 + ⋯ + a k). Assume that A (G) has rank k , so that a k ≠ 0. Can we ever have | a k | = 1 ? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, "Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?" We provide a class of such graphs. Furthermore, we extend our results to weighted graphs. [ABSTRACT FROM AUTHOR]

Bio-inspired neuromorphic cameras asynchronously record pixel brightness changes and generate sparse event streams. They can capture dynamic scenes with little motion blur and more details in extreme illumination conditions. Due to the multidimensional address-event structure, most existing vision algorithms cannot properly handle asynchronous event streams. While several event representations and processing methods have been developed to address such an issue, they are typically driven by a large number of events, leading to substantial overheads in runtime and memory. In this paper, we propose a new graph representation of the event data and couple it with a Graph Transformer to perform accurate neuromorphic classification. Extensive experiments show that our approach leads to better results and excels at the challenging realistic situations where only a small number of events and limited computational resources are available, paving the way for neuromorphic applications embedded into mobile facilities. • Asynchronous event streams of neuromorphic cameras shape synchronous graph representations. • Graph Transformer effectively learns informative features from event-based graphs. • Temporal-wise connectivity and learning suffer less from the loss of information. • Event-based graphs feature lower runtime and smaller memory footprints. [ABSTRACT FROM AUTHOR]

A matrix A is said to be an integer matrix if all its entries are integers. In this article, we characterize the nonsingular integer matrices with integer eigenvalues in an expressible form of their corresponding inverse matrices. It is proved that a nonsingular integer matrix A has integer eigenvalues if and only if A − 1 can be written as the sum of n rank-one matrices that meet certain requirements. A method for constructing integer matrices with integer eigenvalues using the Hadamard product is also provided. Let S represent an n -tuple of nonnegative integers. If there is an n × n integer matrix A whose spectrum (the collection of eigenvalues) is S , we say that S is realisable by an integer matrix. In Fallat et al. (2005) [7] , the authors posed a conjecture that " there is no simple graph on n ≥ 2 vertices whose Laplacian spectrum is given by (0 , 1 , ... , n − 1)." We provide a characterization of threshold graphs using the spectra of quotient matrices of its G -join graphs. As a consequence, we prove that given any n − 1 positive integers λ 2 , ... , λ n such that λ 2 ≤ ⋯ ≤ λ n , the n -tuple (0 , λ 2 , ... , λ n) is realizable by the Laplacian matrix of a multidigraph. In particular, we show that (0 , 1 , ... , n − 1) can be realizable by the Laplacian matrix of a multidigraph. [ABSTRACT FROM AUTHOR]

Nearest neighbor search (NNS) is one of the main concerns in data stream applications since similarity queries can be used in multiple scenarios. Online NNS is usually performed on a sliding window by lazily scanning every element currently stored in the window. This paper proposes Sliding Window-based Incremental Nearest Neighbors (SWINN), a graph-based online search index algorithm for speeding up NNS in potentially never-ending and dynamic data stream tasks. Our proposal broadens the application of online NNS-based solutions, as even moderately large data buffers become impractical to handle when a naive NNS strategy is selected. SWINN enables efficient handling of large data buffers by using an incremental strategy to build and update a search graph supporting any distance metric. Vertices can be added and removed from the search graph. To keep the graph reliable for search queries, lightweight graph maintenance routines are run. According to experimental results, SWINN is significantly faster than performing a naive complete scan of the data buffer while keeping competitive search recall values. We also apply SWINN to online classification and regression tasks and show that our proposal is effective against popular online machine learning algorithms. • We introduce SWINN, a graph-based nearest neighbor search index for sliding windows. • SWINN works with any distance metric and is robust against concept drift. • SWINN is significantly faster than a linear scan of the data buffer. • Our proposal delivers competitive search recalls compared to the traditional strategy. • SWINN is successfully applied to classification and regression case studies. [ABSTRACT FROM AUTHOR]

Let G be a graph and h : E (G) → [ 0 , 1 ] be a function. If g (x) ≤ ∑ e ∋ x h (e) ≤ f (x) for every x ∈ V (G) , then we call a graph F h with vertex set V (G) and edge set E h a fractional (g , f) -factor of G with indicator function h , where e ∋ x means the edge e incident to vertex x in G and E h = { e : e ∈ E (G) , h (e) > 0 }. We say that G admits property E (m , n) with respect to fractional (g , f) -factor if for any two sets of independent edges M and N with | M | = m , | N | = n and M ∩ N = 0̸ , G admits a fractional (g , f) -factor F h such that h (e) = 1 for any e ∈ M and h (e) = 0 for any e ∈ N. In this paper, we show a lower bound on binding number which guarantees a graph G to have property E (1 , n) with respect to fractional (g , f) -factor. [ABSTRACT FROM AUTHOR]

In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular for lollipop graphs and generalized barbell graphs. For a matrix A with associated graph G , a new technique utilizing the strong spectral property is introduced, allowing us to construct a matrix A ′ whose graph is obtained from G by appending a clique while arbitrary list of eigenvalues is added to the spectrum. Consequently, many spectra are shown realizable for block graphs. [ABSTRACT FROM AUTHOR]

Broad learning system (BLS) is viewed as an alternative method of deep neural network (DNN). Comparing with DNN, BLS can reach a similar performance with much faster learning speed. BLS contains two kinds of features: the mapped features and the enhancement nodes. The mapped features are obtained by randomly generated connecting weights and biases which are then fine-tuned by sparse autoencoder. But sparse autoencoder, as an unsupervised leaning algorithm, will lose the information of ground truths. The enhancement nodes are formed based on the mapped features with randomly generated weights and biases. However, this random process cannot guarantee the quality of these formed nodes. It means BLS may be unable to learn useful enough representations of original data. Considering the effectiveness of extreme learning machine based autoencoder (ELM-AE), we first propose a novel graph-based ELM-AE (GBEAE) method. Then, a GBEAE-based broad learning system (GBEAE-BLS) is introduced to overcome the disadvantages discussed above. Finally, we use 23 datasets to test the performance of GBEAE-BLS. Experimental results demonstrate that GBEAE-BLS has evident superiorities on these datasets, comparing with related learning algorithms. [ABSTRACT FROM AUTHOR]

A tensegrity graph is a graph with edges labeled as bars, cables and struts. A realization of a tensegrity graph T is a pair (T , p) , where p maps the vertices of T into R d for some d ≥ 1. The realization is globally rigid if any realization (T , q) in R d in which the bars have the same length and the cables and struts are not longer and not shorter, respectively, is an isometric image of (T , p). A tensegrity graph is weakly globally rigid in R d if it has a generic globally rigid realization in R d , and strongly globally rigid in R d if every generic realization in R d is globally rigid. In this paper we give a necessary condition for weak global rigidity in R d and prove that in the d = 1 case the same condition is also sufficient. In particular, our results imply that a tensegrity graph has a generic globally rigid realization in R 1 if and only if it has a generic universally rigid realization in R 1. We also show that recognizing strongly globally rigid tensegrity graphs in R d is co-NP-hard for all d ≥ 1. [ABSTRACT FROM AUTHOR]

• Exploiting Low-rank representation to obtain low-dimensional subspace structure. • LRR improvement by combination of LRR model and manifold regularization. • Using a cluster-based graph to model the manifold structure. • Graph generating using the information of segments in each cluster. • Dictionary learning to achieve highly discriminative features. One of the most important processes on hyperspectral images (HSI) is classification. Extracting low-dimensional subspace structures and obtaining discriminative representations are among the significant goals in the hyperspectral image classification. The low-rank representation (LRR) method obtains low-dimensional data structures; however, it cannot accurately find the intrinsic and discriminative structure in a nonlinear structure. In this paper, the cluster-based graph regularized LRR with dictionary learning (CLRRDL) is proposed for HSI classification. The data manifold structure is combined with the LRR model in the form of manifold regularization to improve LRR. The information of segments in each cluster is used in the proposed method to obtain the graph. Also, the geodesic distance is used to express the graph similarity with the nonlinear structure. In this model, each pixel is expressed as a linear combination of dictionary components. Moreover, rather than working on the entire image, it first clusters the image by considering the pixels in a cluster to be often composed of the same materials and their linear combination to be limited to common components from the dictionary. Then, the proposed method is separately applied to each cluster. Highly discriminative features are obtained using this factor and dictionary learning. The experimental results show that the proposed method performs better compared to other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

The line sum optimization problem asks for a (0 , 1) -matrix minimizing the sum of given functions evaluated at its row and column sums. We show that the uniform problem, with identical row functions and identical column functions, and the monotone problem, over matrices with nonincreasing row and column sums, are polynomial time solvable. [ABSTRACT FROM AUTHOR]

As an emerging technique for supervised learning, broad learning system (BLS) has been proved to have many advantages such as fast learning speed, good generalization, etc. However, it is difficult for BLS to perform well on the dataset which only contains few labeled samples. Because the focus of semi-supervised learning is utilizing the information between labeled and unlabeled samples while the original BLS apparently ignores this significant information. In BLS, the mapped features are obtained by sparse autoencoder which is an unsupervised learning method and the enhancement nodes are formed based on the mapped features. But sparse autoencoder will definitely lose the information of labeled instances. Therefore, we propose a novel semi-supervised BLS (S2-BLS) in this paper. In S2-BLS, inspired by the effectiveness of extreme learning machine based autoencoder (ELM-AE), we first propose semi-supervised ELM-AE (SS-ELM-AE) to obtain the mapped features. Then, the discriminative projecting weights between the ground truths and the transformed features (including the mapping features and the enhancement nodes) are calculated directly. Finally, we adopt some public datasets to verify the learning ability of S2-BLS. And the experimental results illustrate that S2-BLS has an obvious advantage to learn labeled and unlabeled data, comparing with related algorithms. [ABSTRACT FROM AUTHOR]

We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call "scaffolds", obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of "moves" on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs. [ABSTRACT FROM AUTHOR]

We consider the problem of finding a subgraph of a given graph minimizing the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already for bipartite graphs when the functions are convex on one side and concave on the other, we have that when all functions are convex, the problem can be solved in polynomial time for any graph. We also provide polynomial time solutions for bipartite graphs with one side fixed for arbitrary functions, and for arbitrary graphs when all but a fixed number of functions are either nondecreasing or nonincreasing. We note that the general factor problem and the (l,u)-factor problem over a graph are special cases of our problem, as well as the intriguing exact matching problem. The complexity of the problem remains widely open, particularly for arbitrary functions over complete graphs. [ABSTRACT FROM AUTHOR]

Manoeuvrable, responsive satellite constellations that respond to real time events could provide data on demand for time-critical tasks, such as disaster monitoring and relief efforts. The authors demonstrate the feasibility of such a system by expanding on a fully analytical method for designing responsive spacecraft manoeuvres using low thrust propulsion. These manoeuvres are perceived as a graph that enables efficient exploration and optimised selection of favourable routes that achieve mission goals while highlighting resilience and redundancy in the mission's execution. A case study is presented that considers four satellites required to provide flyovers of two targets, with an associated graph of 10,839 possible manoeuvres. Investigation of the graph highlights that a good, but not minimum time, solution can allow the system to perform well, while also providing greater resilience to changes in mission priorities and errors in execution. This analytical approach enables operators to trade-off between a loss of time by using only one satellite versus the disruption of moving multiple satellites for a potentially faster response. The impact of varying mission capabilities, such as using fewer satellites, smaller swath width or less propellant, can be evaluated by reducing the graph without recalculating manoeuvre options. [ABSTRACT FROM AUTHOR]

Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations. [ABSTRACT FROM AUTHOR]

The change in geometric multiplicity, associated with perturbing a particular pair of off-diagonal entries, in a combinatorially symmetric matrix is investigated. First, we focus upon a Hermitian matrix, with a general graph, when the perturbation is Hermitian. This generalizes prior work in case the graph is a tree; the possibilities are much richer. Then, we turn to general matrices over a field. In both cases, by classifying the incident vertices, all possibilities for the change in geometric multiplicity are identified. Then, examples are given to show that each possibility may actually occur. In some instance, the change in geometric multiplicity depends upon the numerical value of the perturbation, and it matters that both off-diagonal entries change. However, in most instances, the change is qualitatively determined. [ABSTRACT FROM AUTHOR]

Several topological indices (second Zagreb index, augmented Zagreb index, atom-bond connectivity index etc.) are possibly the graph-based molecular structure descriptors most widely applied in chemistry and pharmacology. One important aspect in the study of topological indices is their discrimination ability. In view of this, the exponential vertex-degree-based topological index was introduced in the literature. Cruz et al. (2020) mentioned some open problems on the exponential vertex-degree-based topological indices of trees. In this paper, we solve two open problems on the exponential second Zagreb index and the exponential augmented Zagreb index of trees. Moreover, we present some bounds on the exponential atom-bond connectivity index of graphs and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (G) of a graph G is the largest eigenvalue of its adjacency matrix A (G). In this paper, we first give a sharp lower bound of the spectral radius of a graph G in terms of the degree and the average 2-degree which generalizes a result of Das and Kumar (2003), and then as an application, a sharp upper bound of the energy of G is given which generalizes a result of Liu et al. (2007). [ABSTRACT FROM AUTHOR]

This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called a Base Graph–Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence 2 so that the connected components of the result are isomorphic. Given the Base Graph whose subdivision is isomorphic to each component, and the Connection Graph, which describes how the components overlap, we can, in some cases, provide a construction which can make a graph having such a decomposition. This paper investigates the general phenomenon as well as the special cases in which the connection graph has no more than one edge. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph and A (G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A (G) is nonsingular and 1 λ is an eigenvalue of A (G) whenever λ is an eigenvalue of A (G). Further, if λ and 1 λ have the same multiplicity, for each eigenvalue λ , then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR). [ABSTRACT FROM AUTHOR]

We show that the column sum optimization problem , of finding a (0 , 1) -matrix with prescribed row sums which minimizes the sum of evaluations of given functions at its column sums, can be solved in polynomial time, either when all functions are the same or when all row sums are bounded by any constant. We conjecture that the more general line sum optimization problem , of finding a matrix minimizing the sum of given functions evaluated at its row sums and column sums, can also be solved in polynomial time. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with normalized Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0. Let m G (ρ i) (1 ≤ i ≤ n) be the multiplicity of the normalized Laplacian eigenvalue ρ i of G. In this paper, we give some necessary and sufficient conditions for a connected graph G to have m G (ρ i) = n − 3 for some i. We also discuss about graphs with m G (ρ n − 2) = n − k. Moreover, we completely characterize the connected graphs with m G (ρ n − 1) = n − k for sufficiently large n. [ABSTRACT FROM AUTHOR]

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (M S I) in mathematical chemistry. The eccentric complexity of a graph G is defined to be the number of different eccentricities of its vertices. Hua et al. (2020) mentioned two open problems and a conjecture on the Merrifield–Simmons index and eccentric complexity of graph. In this paper we solve both open problems and a conjecture. Moreover, we generalize some of the results. [ABSTRACT FROM AUTHOR]

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m (G). It was shown by Saks and Werman that m (K n) = n − b (n) , where b (n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n − b (n) ≤ m (G) ≤ n − 1. We show that for any tree T on an even number of vertices we have m (T) = n − 1 , and that for any tree T on an odd number of vertices, we have n − 65 ≤ m (T) ≤ n − 2. Our proof uses results about the weighted version of the problem for K n , which may be of independent interest. We also exhibit a sequence G n of graphs with m (G n) = n − b (n) such that G n has O (n b (n)) edges and n vertices. [ABSTRACT FROM AUTHOR]

On a graph G = (V , E) , we call S ⊂ V resolving if ∀ u , v ∈ V with u ≠ v , ∃ w ∈ V such that d (u , w) ≠ d (v , w). The smallest possible cardinality of S is called the metric dimension, computing which is known to be NP-hard. Solving the metric dimension problem (MDP) and the associated minimal resolving set has many important applications across science and engineering. In this paper, we introduce MaskGNN, a method using a graph neural network (GNN) model to learn the minimal resolving set in a self-supervised manner by optimizing a novel surrogate objective. We provide a construction showing the global minimum of this objective coincides with the solution to the MDP. MaskGNN attains 51%–72% improvement over the best baseline and up to 98% the reward of integer programming in 0.72% of the running time. On this foundation, we introduce Pretraining Position-aware GNNs (PP-GNN) and evaluate on popular benchmark position-based tasks on graphs. PP-GNN's strong results challenge the currently popular paradigm – heuristic-driven anchor-selection – with a new learning-based paradigm — simultaneously learning the metric basis of the graph and pretraining position-aware representations for transferring to downstream position-based tasks. • Formulation of a self-supervised surrogate objective for learning an NP-hard problem • Construction showing the surrogate loss's minimum coincides with the optimal solution • Extensive evaluation and significant improvements over meaningful baselines • Competitive performance to integer programming which is over a hundred times slower • Single-instance pretraining learns transferrable position-aware representations [ABSTRACT FROM AUTHOR]

• Research highlights 1 GTBA-Net is designed to tackle the challenges of complex backgrounds, target-irrelevant noise, and ambiguous boundaries in medical image segmentation. • Research highlights 2 Global Feature Aggregation modules employ an efficient self-attention mechanism to facilitate target localization from complex backgrounds. • Research highlights 3 Graph-based Dynamic Feature Fusion modules utilize graph attention network to suppress target-irrelevant noises and preserve spatial details. • Research highlights 4 Uncertainty-based Boundary Refinement modules introduce an uncertainty quantification strategy and auxiliary loss to discern ambiguous boundaries. Despite the considerable progress achieved by U-Net-based models, medical image segmentation remains a challenging task due to complex backgrounds, irrelevant noises, and ambiguous boundaries. In this study, we present a novel approach called U-shaped Graph- and Transformer-guided Boundary Aware Network (GTBA-Net) to tackle these challenges. GTBA-Net uses the pre-trained ResNet34 as its basic structure, and involves Global Feature Aggregation (GFA) modules for target localization, Graph-based Dynamic Feature Fusion (GDFF) modules for effective noise suppression, and Uncertainty-based Boundary Refinement (UBR) modules for accurate delineation of ambiguous boundaries. The GFA modules employ an efficient self-attention mechanism to facilitate coarse target localization amidst complex backgrounds, without introducing additional computational complexity. The GDFF modules leverage graph attention mechanism to aggregate information hidden among high- and low-level features, effectively suppressing target-irrelevant noises while preserving valuable spatial details. The UBR modules introduce an uncertainty quantification strategy and auxiliary loss to guide the model's focus towards target regions and uncertain "ridges", gradually mitigating boundary uncertainty and ultimately achieving accurate boundary delineation. Comparative experiments on five datasets encompassing diverse modalities (including X-ray, CT, endoscopic procedures, and ultrasound) demonstrate that the proposed GTBA-Net outperforms existing methods in various challenging scenarios. Subsequent ablation studies further demonstrate the efficacy of the GFA, GDFF, and UBR modules in target localization, noise suppression, and ambiguous boundary delineation, respectively. GTBA-Net exhibits substantial potential for extensive application in the field of medical image segmentation, particularly in scenarios involving complex backgrounds, target-irrelevant noises, or ambiguous boundaries. [ABSTRACT FROM AUTHOR]

A P k -factor H of G is a spanning subgraph of G such that each component of H is a path of k vertices, where k ≥ 2. In this paper, we obtain two necessary conditions for P k -factors in the squares and line graphs of trees, respectively. It is shown that the results are best possible in some sense. Furthermore, we give a characterization for a class of trees whose line graphs contain P 4 -factors. • Let T be a tree. If T 2 contains a P k -factor, where k ≥ 4 , then d 1 (T , v) ≥ d k − 1 (T , v) − 3 for any v ∈ V (T). • Let T be a tree. If L (T) contains a P k -factor, where k ≥ 3 , then c 0 (T , v) ≥ c k − 2 (T , v) for any v ∈ V (T). • Give a characterization for a class of trees whose line graphs contain P 4 -factors. [ABSTRACT FROM AUTHOR]

Graph is a widely applied technique to characterize the relationship among data. Due to the excellent ability to handle nonlinear data and extract the useful information contained in base kernels, learning the connecting graph based on multiple kernel learning has been extensively discussed. Many existing algorithms construct the connecting graph based on the optimal kernel which is learned from base kernels. Observing these methods, we find they (1) ignore the local structure of data; (2) cannot assure that the optimal kernel is positive semi-definite; (3) cannot fully utilize the information contained in base kernels. Therefore, we introduce a novel local graph based on multiple kernel learning (LGMKL) in this paper. Specifically, LGMKL is constructed based on the optimal kernel which is automatically learned from base kernels with a nonlinear strategy and the information contained in different base kernels is also utilized in LGMKL. Then an iterative scheme with proven convergence is developed to optimize the objective function of LGMKL. Unlike most MKL-based graph learning methods, LGMKL focuses on the local structure of data. Finally, nine benchmark datasets and two synthetic datasets are adopted to test the performance of LGMKL. Extensive experiments demonstrate the superiorities of the proposed method. [ABSTRACT FROM AUTHOR]

With the development of artificial intelligence, the most representative deep learning has been applied to various fields, which is greatly influencing human society. However, deep neural networks (DNNs) are still a black-box model, and the process how they make decisions internally is still difficult to understand and control. At the same time, DNNs take up more hardware resources, resulting in high energy consumption. Therefore, it is significant to study the characteristics of deep AI models and deeply understand the interactions between parameters within AI models so as to improve the interpretability of DNNs, optimize their structure and increase their computational efficiency. In this paper, we propose a graph-based interpretability method for deep neural networks (GIMDNN). The running parameters of DNNs are modeled as a graph by using a kernel function or the Graph Transformer Networks (GTN), where the nodes of the graph are obtained by dimensional mapping of the parameters of the DNNs, and the edges are calculated by the Gaussian kernel function. The generated graphs are classified by a graph convolutional network (GCN). The association relationship between the adjacent layers and the running mechanism of DNNs are analyzed, and the importance of the parameters of each layer in the DNNs for the final classification result can be obtained. Convolutional neural networks (CNNs) are one of the most representative models in DNNs. The proposed method is experimentally evaluated on the CNNs. The experimental results show that the proposed method can interpret the associations among the weight parameters as well as the correlation between two adjacent layers. Therefore, the DNNs for special tasks, such as portable applications, edge computing, and so on, can be customized, the number of parameters can be reduced. It is valuable to interpret the operation and principle of CNNs. [ABSTRACT FROM AUTHOR]

Reinforcement learning faces the challenge of sparse rewards. Existing research utilizes reward shaping based on graph convolutional neural networks (GCNs) to address this challenge. However, the automatic construction of optimal graph has been a long standing issue. Here we propose Graph Convolution with Topology Refinement for Automatic Reinforcement Learning (GTR), based on the construction of new latent graph to replace the original input graph for more effective reward shaping. It is found from this work that, the most suitable state node can be extracted through the graph entropy. Subsequently we map the original graph to subset of nodes adaptively to form a new and more compact latent graph. Since GTR utilizes trainable projection vectors for projecting all node features into one-dimensional representation, the inter-connections between the nodes of the newly constructed latent graph are consistent with the original ones. The proposed GTR stems from mathematical grounds, and preliminary experiments have shown that the proposed GTR has considerable improvement on Atari benchmark and Mujoco benchmark. Further experiment and ablation analysis have given further supports to this work. [ABSTRACT FROM AUTHOR]

The present paper focuses on the study of the conditions under which the covariance matrix of a multivariate Gaussian distribution is totally positive, paying particular attention to multivariate Gaussian distributions that are Gaussian Markov Random Fields. More specifically, it is proven that, if the graph over which the Gaussian Markov Random Field is defined consists of path graphs and the covariances between adjacent variables on the graph are non-negative, then there always exists a reordering of the variables that renders the resulting covariance matrix totally positive. Moreover, this reordering is identified and some cases for which the conditions for the covariance matrix of a multivariate Gaussian distribution to be totally positive are necessary and sufficient are provided. [ABSTRACT FROM AUTHOR]

The growing accessibility of large-scale protein interaction data demands extensive research to understand cell organization and its functioning at the network level. Bioinformatics and data mining researchers have extensively studied network clustering to examine the structural and operational features of protein protein interaction (PPI) networks. Clustering PPI networks has proven useful in numerous research over the past two decades for identifying functional modules, understanding the roles of previously unknown proteins, and other purposes. Protein complexes represent one of the essential cellular components for creating biological activities. Inferring protein complexes has been made more accessible by experimental approaches. We offer a novel method that integrates the classification model with local topological data, making it more reliable and efficient. This article describes a decision tree classifier based on topological characteristics of the subgraph for mining protein complexes. The proposed graph-based algorithm is an effective and efficient way to identify protein complexes from large-scale PPI networks. The performance of the proposed algorithm is observed in protein–protein interaction networks of yeast and human in the Database of Interacting Proteins (DIP) and the Biological General Repository for Interaction Datasets (BioGRID) using widely accepted benchmark protein complexes from the comprehensive resource of mammalian protein complexes (CORUM) and the comprehensive catalogue of yeast protein complexes (CYC2008). The outcomes demonstrate that our method can outperform the best-performing supervised, semi-supervised, and unsupervised approaches to detecting protein complexes. [Display omitted] • Complex prediction by decision tree and topological characteristics of subgraph(DPMC). • An effective and efficient way to predict complexes from large-scale PPI networks. • Datasets: DIP and BioGRID for PPI network; CORUM and CYC2008 for benchmark complexes • The proposed algorithm outperforms MCODE, DPClus, RNSC, COACH, and ClusterONE. • The proposed algorithm is at par with CART, ClusterSS, BHHO, PCPCRO, PROCOP and ESCC. [ABSTRACT FROM AUTHOR]

This paper aims to discuss the convergence behavior of fixed points for graphical B c -Kannan-contractions using numerical iterations within the structure of graphical extended b -metric spaces. We provide innovative fixed point results using B c -Kannan-contraction and demonstrate that every Kannan contraction is a graphical B c -Kannan, but the converse is not valid in general. The findings, which explain the convergence of fixed points for graph-Kannan mappings, are the first of their kind in state of the art. We give examples using the framework of graphical analysis to show that our important findings are more general than the research that backs them up. Lastly, utilizing our results, we show that a solution to a fourth-order two-point boundary value problem expressing elastic beam deformations exists. [ABSTRACT FROM AUTHOR]

We study here the positionality of trigger strategies Nash equilibria σ ‾ for the N -player SCAR games Γ N (G | s 0 , γ , ε) (with N ≥ 3). Our study is exhaustive with respect to types of graphs G , initial states s 0 and values of N , γ , ε. We conclude that in the majority of cases, profiles σ ‾ are nonpositional. Whenever σ ‾ are positional a key role is played by paths and the ε , γ values (especially whether ε > 0 or not). A crucial concept in our analysis is the state cop number , which is first introduced in the current paper. [ABSTRACT FROM AUTHOR]

The k -independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k -partially walk-regular if the number of closed walks of a given length l ≤ k , rooted at a vertex v , only depends on l. In particular, a distance-regular graph is also k -partially walk-regular for any k. In this paper, we introduce a new family of polynomials obtained from the spectrum of a graph, called minor polynomials. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k -independence number of a k -partially walk-regular graph. With some examples and infinite families of graphs whose bounds are tight, we also show that the odd graph O ℓ with ℓ odd has no 1-perfect code. Moreover, we show that our bound coincides, in general, with the Delsarte's linear programming bound and the Lovász theta number θ , the best ones to our knowledge. In fact, as a byproduct, it is shown that the given bounds also apply for θ and Θ, the Shannon capacity of a graph. Moreover, apart from the possible interest that the minor polynomials can have, our approach has the advantage of yielding closed formulas and, so, allowing asymptotic analysis. [ABSTRACT FROM AUTHOR]

Let m , n , r and k i (1 ≤ i ≤ m) be positive integers with n ≤ m and k 1 ≥ k 2 ≥ ⋯ ≥ k m ≥ 2 r − 1. Let G be a graph, and let H 1 , H 2 , ... , H r be vertex-disjoint n -subgraphs of G. It is verified in this article that every 0 , k 1 + k 2 + ⋯ + k m − n + 1 -graph G includes a subgraph R such that R has a 0 , k i 1 n -factorization orthogonal to every H i , 1 ≤ i ≤ r. [ABSTRACT FROM AUTHOR]

A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of graphical designs under the operation of weak graph product. In addition, we present a family of extremal graphical designs for the hypercube graph. [ABSTRACT FROM AUTHOR]

The connectivity is an important measurement for the fault-tolerance of networks. The structure connectivity and substructure connectivity are two generalizations of the classical connectivity. For a fixed graph H , a set F of subgraphs of G is called an H -structure cut (resp., H -substructure cut) of G , if G − ∪ F ∈ F V (F) is disconnected and every element of F is isomorphic to H (resp., a connected subgraph of H). The H -structure connectivity (resp., H -substructure connectivity) of G , denoted by κ (G ; H) (resp., κ s (G ; H)), is the cardinality of a minimal H -structure cut (resp., H -substructure cut) of G. In this paper, we will establish both κ (S n ; H) and κ s (S n , H) for every H ∈ { K 1 , K 1 , 1 , K 1 , 2 , ... , K 1 , n − 2 , P 4 , P 5 , C 6 } , where S n is the n -dimensional star graph. These results will show that star networks are highly tolerant of structure faults. [ABSTRACT FROM AUTHOR]

This paper is the first from a series of papers that establish a common analogue of the strong component and basilica decompositions for bidirected graphs. A bidirected graph is a graph in which a sign ＋ or − is assigned to each end of each edge, and therefore is a common generalization of digraphs and signed graphs. Unlike digraphs, the reachabilities between vertices by directed trails and paths are not equal in general bidirected graphs. In this paper, we set up an analogue of the strong connectivity theory for bidirected graphs regarding directed trails, motivated by degree-bounded factor theory. We define the new concepts of circular connectivity and circular components as generalizations of the strong connectivity and strong components. In our main theorem, we characterize the inner structure of each circular component; we define a certain binary relation between vertices in terms of the circular connectivity and prove that this relation is an equivalence relation. The nontrivial aspect of this structure arises from directed trails starting and ending with the same sign, and is therefore characteristic to bidirected graphs that are not digraphs. This structure can be considered as an analogue of the general Kotzig–Lovász decomposition, a known canonical decomposition in 1-factor theory. From our main theorem, we also obtain a new result in b -factor theory, namely, a b -factor analogue of the general Kotzig–Lovász decomposition. [ABSTRACT FROM AUTHOR]

In a recent article by Bona and Vince, the authors studied the concept of an assembly tree for a graph motivated by the way that macromolecules assemble. Assembly trees act as record keeping devices for the construction of a given graph from its vertices. In the original article, graphs were to be assembled following an edge gluing rule for the vertices. In this paper, we first define and enumerate assembly trees that are constructed using a connected gluing rule. Following this, we examine how introducing a notion of time to assembly trees changes how they are counted using both gluing rules. [ABSTRACT FROM AUTHOR]

In 1975, P. Erdős proposed the problem of determining the maximum number f (n) of edges in a graph with n vertices in which any two cycles are of different lengths. The sequence (c 1 , c 2 , ... , c n) is the cycle length distribution of a graph G with n vertices, where c i is the number of cycles of length i in G. Let f (a 1 , a 2 , ... , a n) denote the maximum possible number of edges in a graph which satisfies c i ≤ a i , where a i is a nonnegative integer. In 1991, Shi posed the problem of determining f (a 1 , a 2 , ... , a n) which extended the problem due to Erdős. It is clear that f (n) = f (1 , 1 , ... , 1). Let g (n , m) = f (a 1 , a 2 , ... , a n) , where a i = 1 if i ∕ m is an integer, and a i = 0 otherwise. It is clear that f (n) = g (n , 1). We prove that lim inf n → ∞ f (n) − n n ≥ 2 + 40 99 , which is better than the previous bounds 2 (Shi (1988)), and 2 + 7654 19071 (Lai (2017)). We show that lim inf n → ∞ g (n , m) − n n m > 2. 444 , for all even integers m. We make the following conjecture: lim inf n → ∞ f (n) − n n > 2. 444. [ABSTRACT FROM AUTHOR]

We introduce the set G SSP of all simple graphs G with the property that each symmetric matrix corresponding to a graph G ∈ G SSP has the strong spectral property. We find several families of graphs in G SSP and, in particular, characterise the trees in G SSP. [ABSTRACT FROM AUTHOR]

This paper presents an approach for Structure from Motion (SfM) for unorganized complex image sets. To achieve high accuracy and robustness, image triplets are employed and an (approximate) internal camera calibration is assumed to be known. The complexity of an image set is determined by the camera configurations which may include wide as well as weak baselines. Wide baselines occur for instance when terrestrial images and images from small Unmanned Aerial Systems (UAS) are combined. The resulting large (geometric/radiometric) distortions between images make image matching difficult possibly leading to an incomplete result. Weak baselines mean an insufficient distance between cameras compared to the distance of the observed scene and give rise to critical camera configurations. Inappropriate handling of such configurations may lead to various problems in triangulation-based SfM up to total failure. The focus of our approach lies on a complete linking of images even in case of wide or weak baselines. We do not rely on any additional information such as camera configurations, Global Positioning System (GPS) or an Inertial Navigation System (INS). As basis for generating suitable triplets to link the images, an iterative graph-based method is employed formulating image linking as the search for a terminal Steiner minimum tree in the line graph. SIFT (Lowe, 2004) descriptors are embedded into Hamming space for fast image similarity ranking. This is employed to limit the number of pairs to be geometrically verified by a computationally and more complex wide baseline matching method (Mayer et al., 2012). Critical camera configurations which are not suitable for geometric verification are detected by means of classification (Michelini and Mayer, 2019). Additionally, we propose a graph-based approach for the optimization of the hierarchical merging of triplets to efficiently generate larger image subsets. By this means, a complete, 3D reconstruction of the scene is obtained. Experiments demonstrate that the approach is able to produce reliable orientation for large image sets comprising wide as well as weak baseline configurations. [ABSTRACT FROM AUTHOR]

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n (G) vertices and e (G) edges. The elementary cyclic number c (G) of G is defined as c (G) = e (G) − n (G) + ω (G) , where ω (G) is the number of connected components of G. The nullity of G , denoted by η (G) , is the multiplicity of the eigenvalue zero of the adjacency matrix of G. A graph is leaf-free if it has no pendent vertices. In Ma et al. (2016) proved that if G is leaf-free and each component of G contains at least two vertices, then η (G) ≤ 2 c (G) , the equality is attained if and only if G is the union of disjoint cycles, where each cycle has length a multiple of 4. In this paper, we completely characterize all leaf-free graphs with nullity one less than the above upper bound, i.e., η (G) = 2 c (G) − 1. [ABSTRACT FROM AUTHOR]

Let π = { V 1 , ... , V r } be an equitable partition of the vertex set of a directed graph (digraph) X. It is well known that the characteristic polynomial ϕ (X / π , x) of a quotient graph X / π divides that of X , but the remainder part is not well investigated. In this paper, we define a deletion graph X \ π over an equitable partition π , which is a signed directed graph defined for a fixed set of deleting vertices { v ‾ i ∈ V i , i = 1 , ⋯ , r } , and give a similarity transformation exchanging the adjacency matrix A (X) which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph. In fact, we show the result for more general matrices including adjacency matrix of graphs, and as corollaries, we show the followings: (i) a decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part, and (ii) Chen and Chen's result ([4, Theorem 3.1]) on the Ihara-Bartholdi zeta functions on generalized join graphs, and (iii) Teranishi's result [32, Theorem 3.3]. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n and also let ρ (G) be the spectral radius of the graph G. In this paper, we prove that ρ (G) ≤ ρ 2 (G − v k) + 2 d k − 1 for any non-isolated vertex v k of degree d k in G , thus confirming a conjecture posed by Guo et al. (2019) [3]. Moreover, we characterized all connected graphs for which this bound is attained. [ABSTRACT FROM AUTHOR]