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Adaptive transform coding is gaining more and more attention for better mining of image content over fixed transforms such as discrete cosine transform (DCT). As a special case, graph transform learning establishes a novel paradigm for the graph-based transforms. However, there still exists a challenge for graph transform learning-based image codecs design on natural image compression, and graph representation cannot describe regular image samples well over graph-structured data. Therefore, in this article, we propose a cross-channel graph-based transform (CCGBT) for natural color image compression. We observe that neighboring pixels having similar intensities should have similar values in the chroma channels, which means that the prominent structure of the luminance channel is related to the contours of the chrominance channels. A collaborative design of the learned graphs and their corresponding distinctive transforms lies in the assumption that a sufficiently small block can be considered smooth, meanwhile, guaranteeing the compression of the luma and chroma signals at the cost of a small overhead for coding the description of the designed luma graph. In addition, a color image compression framework based on the CCGBT is designed for comparing DCT on the classic JPEG codec. The proposed method benefits from its flexible transform block design on arbitrary sizes to exploit image content better than the fixed transform. The experimental results show that the unified graph-based transform outperforms conventional DCT, while close to discrete wavelet transform on JPEG2000 at high bit-rates. [ABSTRACT FROM AUTHOR]

Let τ (G) and τ G (e) be the number of spanning trees of a connected graph G and the number of spanning trees of G containing edge e. The ratio d G (e) = τ G (e) / τ (G) is called the spanning tree edge density, or simply density, of e. The maximum density dep (G) = max e ∈ E (G) d G (e) is called the spanning tree edge dependence, or simply dependence, of G. In 2016, Kahl made the following two conjectures. (C1) Let p , q be positive integers, p < q. There exists some function f (p , q) such that, if G is the bipartite construction as given in Theorem 3.5 (Theorem 2.1 in the present paper), then t i ≥ f (p , q) for all 2 ≤ i ≤ n implies that dep (G) = p / q. (C2) Let p , q be positive integers, p < q. There exists a planar graph G such that dep (G) = p / q. In this paper, by using combinatorial and electrical network approaches, firstly, we show (C1) is true. Secondly, we disprove (C2) by showing that for any planar graph G , dep (G) > 1 3 . On the other hand, by construction families of planar graphs, we show that, for p / q > 1 2 , there do exist a planar graph G such that dep (G) = p / q. Furthermore, we prove that the second conjecture holds for planar multigraphs. [ABSTRACT FROM AUTHOR]

As for single-label learning, feature selection has become a crucial data pre-processing tool for multi-label learning due to its ability to reduce the feature dimensionality to mitigate the influence of "the curse of dimensionality". In recent years, the multi-label feature selection (MFS) methods based on manifold learning and sparse regression have confirmed their superiority and gained more and more attention than other methods. However, most of these methods only consider exploiting the geometric structure distributed in the data manifold but ignore the geometric structure distributed in the feature manifold, resulting in incomplete information about the learned manifold. Aiming to simultaneously exploit the geometric information of data and feature spaces for feature selection, this paper proposes a novel MFS method named dual-graph based multi-label feature selection (DGMFS). In the framework of DGMFS, two nearest neighbor graphs are constructed to form a dual-graph regularization to explore the geometric structures of both the data manifold and the feature manifold simultaneously. Then, we combined the dual-graph regularization with a non-convex sparse regularization l2,1 − 2-norm to obtain a more sparse solution for the weight matrix, which can better deal with the redundancy features. Finally, we adopt the EM strategy to design an iterative updating optimization algorithm for DGMFS and provide the convergence analysis of the optimization algorithm. Extensive experimental results on ten benchmark multi-label data sets have shown that DGMFS is more effective than several state-of-the-art MFS methods. [ABSTRACT FROM AUTHOR]

Relating image contours and regions and their attributes according to connectivity based on incidence or adjacency is a crucial task in numerous applications in the fields of image processing, computer vision and pattern recognition. In this paper, the crucial incidence topological information of 2-dimensional images is extracted in an efficient manner through the computation of a new structure called the HomDuRAG of an image; that is, the dual graph of the HomRAG (a topologically consistent extended version of the classical RAG). These representations are derived from the two traditional self-dual square grids (in which physical pixels play the role of 2-dimensional cells) and encapsulate the whole set of topological features and relations between the three types of objects embedded in a digital image: 2-dimensional (regions), 1-dimensional (contours) and 0-dimensional objects (crosses). Here, a first version of a fully parallel algorithm to compute this new representation is presented, whose timing complexity order (in the worst case and supposing one processing element per 0-cell) is O (l o g (M × N)) , M and N being the height and width of the image. Efficient implementations of this parallel algorithm would allow images to be processed in real time, as well as permit us to uncover fast algorithms for contour detection and segmentation, opening new perspectives within the image processing field. [ABSTRACT FROM AUTHOR]

We find all polyhedral graphs such that their complements are still polyhedral. These turn out to be all self-complementary. [ABSTRACT FROM AUTHOR]

The dual of a polyhedron is a polyhedron—or in graph‐theoretical terms: the dual of a 3‐connected plane graph is a 3‐connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity of the dual on higher surfaces. Are the duals of 3‐connected embedded graphs of higher genus 3‐connected, too? If not, which connectivity guarantees 3‐connectedness of the dual? In this article, we give answers to some of these and related questions. We prove that there is no connectivity that guarantees the 3‐connectedness or 2‐connectedness of the dual for every genus, and give upper bounds for the minimum genus for which (with c>2 $c\gt 2$) a c‐connected embedded graph with a dual that has a 1‐ or 2‐cut can occur. We prove that already on the torus, we need 6‐connectedness to guarantee 3‐connectedness of the dual and 4‐connectedness to guarantee 2‐connectedness of the dual. In the last section, we answer a related question by Plummer and Zha on orientable embeddings of highly connected noncomplete graphs. [ABSTRACT FROM AUTHOR]

The rapid development of natural language processing (NLP) technologies has enabled the emergence of legal intelligence assistance systems, with legal charge prediction (LCP) being a critical technology. The automatic LCP aims to determine the final charges based on fact descriptions of criminal cases. LCP assists human judges in managing workloads and improving efficiency, provides accessible legal guidance for individuals, and supports enterprises in litigation financing and compliance monitoring. However, distinguishing between confusing charges in real-world judicial practice remains a significant challenge. Most exist works cannot effectively capture complex relationships and discern subtle differences in fact descriptions while ignoring the legal schematic knowledge. In order to improve confusing LCP performance, we propose a novel knowledge-aware model for legal charge prediction that leverages Graph Neural Networks (GNNs) to capture complex relationships within criminal case descriptions. Specifically, the model constructs structural and semantic graphs from fact descriptions and integrates information from both through a dual-graph interaction process. A legal knowledge transformer generates key knowledge representations at schema and charge levels, while a knowledge matching network incorporates hierarchical charge knowledge into facts. Besides, we also propose two real-world datasets namely Criminal-All and Criminal-Confusing, containing 203 different charges and 86 confusing charges, respectively. To the best of our knowledge, these datasets are the first well-organized datasets for confusing LCP task. Experimental results demonstrate that the proposed model outperforms baselines and significantly improves the distinction of confusing charges, providing valuable support for intelligent legal judgment systems. • A legal schematic knowledge-aware model for confusing charges. • Dual-graph interaction to integrate information from structural and semantic graphs. • A hierarchical transformer to obtain legal knowledge representations. • A deep matching network for incorporating domain knowledge into the fact. • Our model improvement outperforms baselines and achieves a significant. [ABSTRACT FROM AUTHOR]

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed.

Relating image contours and regions and their attributes according to connectivity based on incidence or adjacency is a crucial task in numerous applications in the fields of image processing, computer vision and pattern recognition. In this paper, the crucial incidence topological information of 2-dimensional images is extracted in an efficient manner through the computation of a new structure called the HomDuRAG of an image; that is, the dual graph of the HomRAG (a topologically consistent extended version of the classical RAG). These representations are derived from the two traditional self-dual square grids (in which physical pixels play the role of 2-dimensional cells) and encapsulate the whole set of topological features and relations between the three types of objects embedded in a digital image: 2-dimensional (regions), 1-dimensional (contours) and 0-dimensional objects (crosses). Here, a first version of a fully parallel algorithm to compute this new representation is presented, whose timing complexity order (in the worst case and supposing one processing element per 0-cell) is O(log(M×N)) , M and N being the height and width of the image. Efficient implementations of this parallel algorithm would allow images to be processed in real time, as well as permit us to uncover fast algorithms for contour detection and segmentation, opening new perspectives within the image processing field.

Let G be a plane graph with a set of faces F (G) and ψ : V (G) → { 1 , 2 , ... , k } be a (not necessarily proper) vertex k -coloring. ψ is called polychromatic if all k colors appear on the vertices of its boundary for each face (also the outer-face) f ∈ F (G). The polychromatic number of G is the largest number of colors k such that G admits a polychromatic k -coloring. In the paper, we consider a class of plane graphs with minimum face size g , in each of which every pair of faces have at most t common incident vertices, and show that the polychromatic number for each G ∈ is at least g − t − 2. When g ≥ 4 t + 4 , the result improves a known lower bound. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization methods have been widely used in many applications in recent years. However, the clustering performances of such methods may deteriorate dramatically in the presence of non-Gaussian noise or outliers. To overcome this problem, in this paper, we propose correntropy-based dual graph regularized NMF with LP smoothness (CDNMFS) for data representation. Specifically, we employ correntropy instead of the Euclidean norm to measure the incurred reconstruction error. Furthermore, we explore the geometric structures of both the input data and the feature space and impose an Lp norm constraint to obtain an accurate solution. In addition, we introduce an efficient optimization scheme for the proposed model and present its convergence analysis. Experimental results on several image datasets demonstrate the superiority of the proposed CDNMFS method. [ABSTRACT FROM AUTHOR]

Urban street profiling is the spatio-temporal pattern discovery of street-level urban areas, which plays a vital role in understanding urban structures and dynamics. Due to the natural topology and various geographic characteristics on the streets, it is necessary to combine multi-dimensional spatio-temporal information to understand different profiles of streets. This research aims to develop a street profiling framework according to the coupled characteristics of streets. At the start, a bidirected dual graph and a spatial weighted graph embedding method were used to solve the street representation. Then, the street profiles can be extracted by clustering embedding vectors of streets and feature importance analysis. As the case study, we employed the bike trajectories and street view images in Xiamen, China to depict the geographic attributes of streets. The results can reveal nine spatio-temporal street profiles from the biking perspective, including three spatial distribution patterns and two spatial semantic patterns. Urban streets in the study area show a significant hierarchical pattern because of locations and the spatial lags of the biking behaviors. Meanwhile, the spatio-temporal characteristics of biking behaviors are the main factors of street profiles, though the street environment attributes participate in over half the number of profile types. We further evaluated the profiling ability of the proposed framework and the importance of urban street profiles using coupled characteristics. Overall, this study explored the profiling method for coupling static and dynamic characteristics of urban streets. The profiling results also help understand street usage and experiences by bikers, which have a practical value on the human-oriented classification of streets and further urban development from a geographic view. • Urban street profiling by the coupled spatio-temporal patterns and biking activities. • Bidirected dual graph was employed to couple multi-dimensional spatial attributes. • Spatial weighted GraphMAE was improved to train the embedding vectors of streets. • Nine types of urban street profiles were found with three spatial distribution. [ABSTRACT FROM AUTHOR]

Distance-hereditary graphs can be characterized by every cycle of length at least 5 having crossing chords. This makes distance-hereditary graphs susceptible to dualizing, using the common extension of geometric face/vertex planar graph duality to cycle/cutset duality as in abstract matroidal duality. The resulting “DH* graphs” are characterized and then analyzed in terms of connectivity. These results are used in a special case of plane-embedded graphs to justify viewing DH* graphs as the duals of distance-hereditary graphs.

Drainage pattern recognition (DPR) is a classic and challenging problem in hydrographic system analysis, topographical knowledge mining, and map generalization. An outstanding issue for traditional DPR methods is that the rules used to extract patterns based on certain geometric measures are limited, not accessing the effects of manual recognition. In this study, a graph convolutional network (GCN) was introduced for DPR. First, a dual graph of drainage was built based on the channel connection and hierarchical structure after constructing typical sample data. Second, its features were extracted as inputs of the GCN from three scales, namely, global unity at a macroscale, hierarchical connectivity at a mesoscale, and local equilibrium at a microscale. Finally, the model architecture based on the GCN was designed for DPR. Typical pattern samples (i.e. dendritic, distributary, parallel, skeleton, and rectangular drainage) from OpenStreetMap and USGS were used to implement the training and testing of the model, respectively. The results show that our approach outperforms other machine learning methods, including convolutional neural network, with an accuracy of 85.0%. In summary, the GCN has considerable potential for DPR and a wide scope for further improvement.

Abstract Computing $$k\text {-}core$$ k-core s is a fundamental and important graph problem, which can be applied in many areas, such as community detection, network visualization, and network topology analysis. Due to the complex relationship between different entities, dual graph widely exists in the applications. A dual graph contains a physical graph and a conceptual graph, both of which have the same vertex set. Given that there exist no previous studies on the $$k\text {-}core$$ k-core in dual graphs, we formulate a k -connected core ($$k\text {-}CCO$$ k-CCO ) model in dual graphs. A $$k\text {-}CCO$$ k-CCO is a $$k\text {-}core$$ k-core in the conceptual graph, and also connected in the physical graph. Given a dual graph and an integer k, we propose a polynomial time algorithm for computing all $$k\text {-}CCO$$ k-CCO s. We also propose three algorithms for computing all maximum-connected cores ($$MCCO$$ MCCO ), which are the existing $$k\text {-}CCO$$ k-CCO s such that a $$(k+1)$$ (k+1) -$$CCO$$ CCO does not exist. We further study a subgraph search problem, which is computing a $$k\text {-}CCO$$ k-CCO that contains a set of query vertices. We propose an index-based approach to efficiently answer the query for any given parameter k. We conduct extensive experiments on six real-world datasets and four synthetic datasets. The experimental results demonstrate the effectiveness and efficiency of our proposed algorithms.

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After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene's stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of C40, C60 and C80. This paper suggests a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations (Sure et al. in Phys Chem Chem Phys 19:14296–14305, 2017). We apply our method to classify all isomers of C60 and give an example of two different cospectral isomers of C44. Calculations are done with our own Python scripts available at (Bille et al. in Fullerene database and classification software, https://www.uni-ulm.de/mawi/mawi-stochastik/forschung/fullerene-database/, 2020). The only input for our algorithm is the vector of positions of pentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software package Fullerene (Schwerdtfeger et al. in J Comput Chem 34:1508–1526, 2013). [ABSTRACT FROM AUTHOR]

In the data mining of road networks, trajectory clustering of moving objects is of particular interest for its practical importance in many applications. Most of the existing approaches to this problem are based on distance measurement, and suffer from several performance limitations including inaccurate clustering, expensive computation, and incompetency to handle high dimensional trajectory data. This paper investigates the complex network theory and explores its application to trajectory clustering in road networks to address these issues. Specifically, we model a road network as a dual graph, which facilitates an effective transformation of the clustering problem from sub-trajectories in the road network to nodes in the complex network. Based on this model, we design a label-based trajectory clustering algorithm, referred to as LBTC, to capture and characterize the essence of similarity between nodes. For the evaluation of clustering performance, we establish a clustering criterion based on the classical Davies-Bouldin Index (DB), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) to maximize inter-cluster separation and intra-cluster homogeneity. The clustering accuracy and performance superiority of the proposed algorithm are illustrated by extensive simulations on both synthetic and real-world dataset in comparison with existing algorithms. [ABSTRACT FROM AUTHOR]

It is shown that in the three-dimensional Euclidean space a convex pentagonal prism is not a primitive polyhedron. Some properties of primitive polyhedrons are investigated and the associated dual graphs are considered. [ABSTRACT FROM AUTHOR]

A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes. [ABSTRACT FROM AUTHOR]

Motivated in part by Propp's intruded Aztec diamond regions, we consider hexagonal regions out of which two horizontal chains of triangular holes (called ferns) are removed, so that the chains are at the same height, and are attached to the boundary. By contrast with the intruded Aztec diamonds (whose numbers of domino tilings contain some large prime factors in their factorization), the number of lozenge tilings of our doubly-intruded hexagons turns out to be given by simple product formulas in which all factors are linear in the parameters. We present in fact q -versions of these formulas, which enumerate the corresponding plane-partitions-like structures by their volume. We also pose some natural statistical mechanics questions suggested by our set-up, which should be possible to tackle using our formulas. [ABSTRACT FROM AUTHOR]

• We enumerate spanning trees with a Kekule structure in the linear hexagonal chains on the plane. • We enumerate spanning trees with a Kekule structure in the linear hexagonal chains on the cylinder. • We enumerate spanning trees with a Kekule structure in the linear hexagonal chains on the Mobius strip. In chemical graph theory, many topological indices of the hexagonal chains, for instance, the energy, Wiener and Kirchhoff indices, and numbers of Kekulé structures and spanning trees, and so on, have been studied extensively. We enumerate spanning trees with a Kekulé structure in the linear hexagonal chains on the plane, cylinder and Möbius strip, respectively. [ABSTRACT FROM AUTHOR]

Recent multi-data fusion techniques have inspired multi-view clustering study, which has become a popular topic in machine learning. One successful way of solving multi-view clustering is nonnegative matrix factorization. This paper proposes an auto weighted robust dual graph nonnegative matrix factorization (ARDNMF), where ℓ 2 , 1 norm is adopted to make ARDNMF robust as data contain noise and outliers. Through dual graphs, structure information in data space and feature space is considered. Moreover, an auto weighted assignment method is introduced in ARDNMF to assign appropriate weights to views. Effective updating rules are given. Experiments on real datasets show ARDNMF superiors to state-of-the-art algorithms in terms of clustering results. • The application of the ℓ 2 , 1 norm makes ARDNMF robust to noise and outliers. • The structure information of each view data is utilized sufficiently to avoid information loss. • An auto weighted strategy is employed, which makes ARDNMF find complementary information among different views flexibly. [ABSTRACT FROM AUTHOR]