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Multigraphs with heterogeneous views present one of the most challenging obstacles to classification tasks due to their complexity. Several works based on feature selection have been recently proposed to disentangle the problem of multigraph heterogeneity. However, such techniques have major drawbacks. First , the bulk of such works lies in the vectorization and the flattening operations, failing to preserve and exploit the rich topological properties of the multigraph. Second , they learn the classification process in a dichotomized manner where the cascaded learning steps are pieced in together independently. Hence, such architectures are inherently agnostic to the cumulative estimation error from step to step. To overcome these drawbacks, we introduce MICNet (multigraph integration and classifier network), the first end-to-end graph neural network based model for multigraph classification. First, we learn a single-view graph representation of a heterogeneous multigraph using a GNN based integration model. The integration process in our model helps tease apart the heterogeneity across the different views of the multigraph by generating a subject-specific graph template while preserving its geometrical and topological properties conserving the node-wise information while reducing the size of the graph (i.e., number of views). Second, we classify each integrated template using a geometric deep learning block which enables us to grasp the salient graph features. We train, in end-to-end fashion, these two blocks using a single objective function to optimize the classification performance. We evaluate our MICNet in gender classification using brain multigraphs derived from different cortical measures. We demonstrate that our MICNet significantly outperformed its variants thereby showing its great potential in multigraph classification. [ABSTRACT FROM AUTHOR]

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each k≥4 that any Kk‐minor‐free multigraph of maximum degree Δ has strong chromatic index at most 32(k−2)Δ. We present a construction certifying that if true the conjecture is asymptotically sharp as Δ→∞. In support of the conjecture, we show it in the case k=4 and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for Kk‐minor‐free simple graphs, the problem of strong edge‐colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For k≥5, we also show that Kk‐minor‐free multigraphs of edge‐diameter at most 2 have strong clique number at most (k−12)Δ. [ABSTRACT FROM AUTHOR]

Graph node clustering methods, which aim to partition graph vertices into several disjoint groups of data with similar features, are usually fulfilled based on topological structural similarity of nodes, such as connectivity between vertices or neighborhood similarity of them. However, the attribute-based clustering is recently challenging to data clustering. The present paper contributes to considering a novel data clustering algorithm, called FBC-Cluster, based on fuzzy multigraphs in terms of both structural and attribute similarities. In the proposed algorithm, attribute similarity is achieved through m-polar fuzzy T-equivalences among alternatives (objects) and structural similarity is defined based on a new similarity measurement, called behavioral similarity index, using closed neighborhood in the attributed clusters. The output of the proposed clustering algorithm includes two main categories, namely certain and possible clusters, based on threshold level β given on the behavioral similarity index. A numerical example is discussed to demonstrate the performance of the designed clustering algorithm. The quality of resultant clusters is also evaluated through density and entropy functions. [ABSTRACT FROM AUTHOR]

The main challenge of cross-modal retrieval is how to efficiently realize cross-modal semantic alignment and reduce the heterogeneity gap. However, existing approaches either ignore the multigrained semantic knowledge learning from different modalities, or fail to learn consistent relation distributions of semantic details in multimodal instances. To this end, this article proposes a novel end-to-end cross-modal representation method, termed as deep multigraph-based hierarchical enhanced semantic representation (MG-HESR). This method is an integration of MG-HESR with cross-modal adversarial learning, which captures multigrained semantic knowledge from cross-modal samples and realizes fine-grained semantic relation distribution alignment, and then generates modalities-invariant representations in a common subspace. To evaluate the performance, extensive experiments are conducted on four benchmarks. The experimental results show that our method is superior than the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Methods to learn informative representations of road networks constitute an important prerequisite to solve multiple traffic analysis tasks with data-driven models. Most existing studies are only developed from a topology structure or traffic attribute perspective, and the resulting representations are biased and cannot fully capture the complex traffic flow patterns that are attributed to human mobility in road networks. Moreover, real-world road networks usually contain millions of segments, which poses a great challenge regarding the memory usage and computational efficiency of existing methods. Consequently, we proposed a novel multiview representation learning framework for large-scale urban road networks to simultaneously preserve topological and human mobility information. First, the road network was modeled as a multigraph, and a multiview random walk method was developed to capture the structure function of the road network from a topology-aware graph and vehicle transfer pattern from a mobility-aware graph. In this process, a large-scale road network organization method was established to improve the random walk algorithm efficiency. Finally, word2vec was applied to learn representations based on sequences that were generated by the multiview random walk. In the experiment, two real-world datasets were used to demonstrate the superior performance of our framework through a comparative analysis. [ABSTRACT FROM AUTHOR]

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Let H be a graph possibly with loops and G a multigraph without loops. G is said to be H-colored if there exists a function c: E(G) → V(H). A cycle ( v 0 , e 0 , v 1 , e 1 , ... , e k - 1 , v k = v 0 ) in G, where e i = v i v i + 1 for every i in {0, ... , k - 1 }, is an H-cycle if and only if (c( e 0 ), a 0 , c( e 1 ), ... , c( e k - 2 ), a k - 2 , c( e k - 1 ), a k - 1 , c( e 0 )) is a walk in H, with a i = c( e i )c( e i + 1 ) for every i in {0, ... , k - 1 } (indices modulo k). If H is a complete graph without loops, an H-walk is called properly colored walk. The problem of check whether an edge-colored graph G contains a properly colored cycle was studied first by Grossman and Häggkvist. Subsequently Yeo gave a sufficient condition which guarantee the existence of a properly colored cycle. In this paper we will extend Yeo's result for the case where stronger requirements are enforced for a properly colored cycle to be eligible, based on the adjacencies of a graph whose vertices are in bijection with the colors. The main result establishes that if H is a graph without loops and G is an H-colored multigraph such that (1) H and G have no isolated vertices, (2) G has no H-cycles and (3) for every x in V(G), G x is a complete k x -partite graph for some k x in N . Then there exists a vertex z in V(G) such that every connected component D of G - z satisfies that {e ∈ E(G) : e = z u for some u in V(D)} is an independent set in G z (where for w in V(G), G w is an associated graph to the vertex w, respect to the H-coloring of G). [ABSTRACT FROM AUTHOR]

Low-rank-tensor-approximation(LRTA)-based hyperspectral image and hyperspectral imagery (HSI) restoration has drawn increasing attention. However, most of the methods construct a hidden low-rank tensor by utilizing the nonlocal self-similarity (NLSS) and global spectral correlation (GSC) inherited by HSIs. Although achieving state-of-the-art (SOTA) restoration performance, NLSS and GSC have limitations. NLSS is introduced from natural image denoising to remove spatially independent identically distributed (i.i.d.) Gaussian and impulse noise, while GSC, which is naturally possessed by HSIs, is adopted to maintain the spectral integrity and remove spectrally, i.i.d., degradations. Therefore, NLSS and GSC may not be successfully used for complex HSI restoration tasks, such as destriping, cloud removal, and recovery of atmospheric absorption bands. To solve the issue, borrowing the idea from manifold learning, the geometry information characterized by proximity relationship, is integrated with the LRTA to solve the above issue, named multigraph-based LRTA (MGLRTA). Different from most of the existing methods, the proposed MGLRTA directly models an HSI as a low-rank tensor and efficiently explores the extra proximity information on the defined graphs that are not only inherited by the low-rank constraints but also naturally possessed in HSIs. A well-posed iterative algorithm is designed to solve the restoration problem. Experimental results on different datasets that cover several severe degradation scenarios demonstrate that the proposed MGLRTA outperforms the SOTA HSI restoration methods. [ABSTRACT FROM AUTHOR]

One of the Combinatorial problems in Operations Research is Travelling Salesman Problem (TSP) with NP-hard in nature. TSP has several application as a parent or sub-problem such as logistics, manufacturing, planning. DNA sequencing etc. Whereas stochastic transportation problem is based on random parameter. In this paper, we have proposed a multi-graph stochastic TSP with 7 nodes and the weights are non-deterministic in nature. Two- optimal method has been used to solve the proposed problem to obtain the optimal solution in terms of minimum distance. A real-li fe numerical problem has also been considered for testing the feasibility of the method on the proposed problem. The method can be used to large- scale problem of same nature of TSP in stochastic environment. [ABSTRACT FROM AUTHOR]

The (real) minimum semidefinite rank of a signed graph is the minimum rank among all real symmetric positive semidefinite matrices associated to the graph and having the given sign pattern. We give a new lower bound for the minimum semidefinite rank of a signed multigraph and show it equals a new upper bound for signed complete multigraphs. This allows a complete characterization of signed multigraphs with minimum semidefinite rank two. We also determine the minimum semidefinite rank of all signed wheel graphs. [ABSTRACT FROM AUTHOR]

As one of the most complex social problems around the world, crime may bring the risk of dying or losing property to the public if not handled properly. Crime prediction which aims at predicting crime incidents before they happen is of great importance to fight against crime. Previous studies are concerned primarily with day-level crime prediction and have certain limitations on modeling complex spatial–temporal-categorical dependency contained in the criminal activities as well as utilizing external factors to facilitate the forecast. In this paper, we develop a novel Neural Attentive framework for Hour-level Crime prediction (NAHC) to cope with these challenges. Specifically, we first adopt the priori knowledge-based data enhancement strategy to alleviate the zero-inflated issue raised in hour-level settings. Then, multi-graph convolutional networks are applied to capture spatial dependency from different aspects. After that, we integrate gated recurrent units with a temporal attention mechanism to jointly address temporal dependency and capture time-sensitive external factors. A categorical attention mechanism is proposed for dealing with categorical dependency and finally a fully connected network is utilized to generate the final prediction results. Extensive experiments on two real-world crime datasets demonstrate the effectiveness of our framework over the state-of-the-art comparing methods. [ABSTRACT FROM AUTHOR]

The irregularity strength of a graph G, s(G), is the least k admitting a {1,2,...,k}‐weighting of the edges of G assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of G via multiplying some of its edges. The most well‐known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant C such that s(G)≤nd+C for each d‐regular graph G with n vertices and d≥2 (while a straightforward counting argument yields s(G)≥n+d−1d). The best known results towards this imply that s(G)≤6⌈nd⌉ for every d‐regular graph G with n vertices and d≥2, while s(G)≤(4+o(1))nd+4 if d≥n0.5lnn. We show that the conjecture of Faudree and Lehel holds asymptotically in the cases when d is neither very small nor very close to n. We in particular prove that for large enough n and d∈[ln8n,nln3n], s(G)≤nd(1+8lnn), and thereby we show that s(G)=nd(1+o(1)) then. We moreover prove the latter to hold already when d∈[ln1+εn,nlnεn], where ε is an arbitrary positive constant. [ABSTRACT FROM AUTHOR]

Let G=(V(G),E(G)) be a multigraph with maximum degree Δ(G), chromatic index χ′(G), and total chromatic number χ″(G). The total coloring conjecture proposed by Behzad and Vizing, independently, states that χ″(G)≤Δ(G)+μ(G)+1 for a multigraph G, where μ(G) is the multiplicity of G. Moreover, Goldberg conjectured that χ″(G)=χ′(G) if χ′(G)≥Δ(G)+3 and noticed the conjecture holds when G is an edge‐chromatic critical graph. By assuming the Goldberg–Seymour conjecture, we show that χ″(G)=χ′(G) if χ′(G)≥max{Δ(G)+2,∣V(G)∣+1} in this note. Consequently, χ″(G)=χ′(G) if χ′(G)≥Δ(G)+2 and G has a spanning edge‐chromatic critical subgraph. [ABSTRACT FROM AUTHOR]

An (n , s , q) -graph is an n -vertex multigraph in which every s -set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n , s , q) -graphs. We give a general lower bound construction for this problem for many pairs (s , q) , which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the (s , q) = (4 , 6 a + 3) case of the problem (for a ≥ 2); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for 'sparse' multigraphs (i.e. when q ≤ 2 ( s 2 )). Finally we introduce some tools that are likely to be useful for attacking the problem in general. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) have witnessed an unprecedented proliferation in tackling several problems in computer vision, computer-aided diagnosis and related fields. While prior studies have focused on boosting the model accuracy, quantifying the reproducibility of the most discriminative features identified by GNNs is still an intact problem that yields concerns about their reliability in clinical applications in particular. Specifically, the reproducibility of biological markers across clinical datasets and distribution shifts across classes (e.g., healthy and disordered brains) is of paramount importance in revealing the underpinning mechanisms of diseases as well as propelling the development of personalized treatment. Motivated by these issues, we propose, for the first time, reproducibility-based GNN selection (RG-Select), a framework for GNN reproducibility assessment via the quantification of the most discriminative features (i.e., biomarkers) shared between different models. To ascertain the soundness of our framework, the reproducibility assessment embraces variations of different factors such as training strategies and data perturbations. Despite these challenges, our framework successfully yielded replicable conclusions across different training strategies and various clinical datasets. Our findings could thus pave the way for the development of biomarker trustworthiness and reliability assessment methods for computer-aided diagnosis and prognosis tasks. RG-Select code is available on GitHub at https://github.com/basiralab/RG-Select. [ABSTRACT FROM AUTHOR]

The presented herb recommendation system aims to analyze the patients' symptoms and recommends a set of herbs as the prescription to treat diseases. In addition to symptoms, the patients' personal properties and induced diagnoses are also essential for treatment making. Specifically, for different age groups, the treatments are different. However, the existing studies only use symptoms to represent patients and ignore the patients' multidimensional features modeling. Thus, these models are insufficiently personalized. Meanwhile, most of these existing herb recommendation models based on graphs have not distinguished the effects of different node types. To address the above limitations, we propose a model named Patient-Oriented Multi-Graph Convolutional Network-based Herb Recommendation system (PMGCN). The prediction model contains two effective modules, patient portraits modeling and herb interactions modeling, to learn representations for patients and enhance herb interactions. First, we depict the patient portrait to enrich the individualized features. To distinguish personal properties, symptoms, and diagnoses, we adopt the type-aware attention mechanism, thereby improving the accuracy of personalized herb recommendation. Next, we build two herb-interaction graphs and design type-aware multigraph convolution networks to capture the interactions of herbs and patient features. In this way, our model emphasizes the impact of the patient portrait on diagnosis induction and herb selection. Experimental studies demonstrate that our method outperforms the compared methods and confirms the significance of patient portraits. In conclusion, this research proposes type-aware multigraph convolution networks and adds patient portraits modeling to simulate TCM prescriptions making. [ABSTRACT FROM AUTHOR]

With the rapid development of online social recommendation system, substantial methods have been proposed. Unlike traditional recommendation system, social recommendation performs by integrating social relationship features, where there are two major challenges, i.e., early summarization and data sparsity. Thus far, they have not been solved effectively. In this article, we propose a novel social recommendation approach, namely Multi-Graph Heterogeneous Interaction Fusion (MG-HIF), to solve these two problems. Our basic idea is to fuse heterogeneous interaction features from multi-graphs, i.e., user--item bipartite graph and social relation network, to improve the vertex representation learning. A meta-path cross-fusion model is proposed to fuse multi-hop heterogeneous interaction features via discrete cross-correlations. Based on that, a social relation GAN is developed to explore latent friendships of each user. We further fuse representations from two graphs by a novel multi-graph information fusion strategy with attention mechanism. To the best of our knowledge, this is the first work to combine meta-path with social relation representation. To evaluate the performance of MG-HIF, we compare MG-HIF with seven states of the art over four benchmark datasets. The experimental results show that MG-HIF achieves better performance. [ABSTRACT FROM AUTHOR]

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture, that may be attributed to Chartrand et al., is that almost every graph G can be turned into a locally irregular multigraph M, i.e., with no two adjacent vertices having the same degree, by replacing each of its edges by at most three parallel edges. In other words, for almost every graph G there should be a locally irregular multigraph M with the same adjacencies and having a relatively small number of edges. The 1–2–3 Conjecture, if true, would indeed imply that there is such an M with | E (M) | ≤ 3 | E (G) | . In this work, we study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. In other words, given a graph G, we are looking for a locally irregular multigraph M ∗ with the smallest number of edges possible that can be obtained from G by multiplying edges. This problem is actually quite different from the 1–2–3 Conjecture, as we prove that there is no absolute constant k such that M ∗ can always be obtained from G by replacing each edge with at most k parallel edges. We investigate several aspects of this problem, covering algorithmic and combinatorial aspects. In particular, we prove that the problem of designing proper labellings with minimum label sum is N P -hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that for almost every connected graph G there should be a proper labelling with label sum at most 2|E(G)|, which we verify for several classes of graphs. [ABSTRACT FROM AUTHOR]

A star edge-coloring of a multigraph G is a proper edge-coloring of G such that no path or cycle of length four is bi-colored. The star chromatic index of G is the minimum number of colors needed to guarantee that G admits a star edge-coloring. The list star chromatic index of G is the smallest integer k such that for any k -uniform list assignment L for the set of edges, G has a star edge-coloring from L. Dvořák, Mohar and Šámal proved that every subcubic multigraph has star chromatic index at most 7, and conjectured that 7 can be further improved to 6. Lužar, Mockovčiaková and Soták strengthened the result of Dvořák, Mohar and Šámal by showing that every subcubic multigraph has list star chromatic index at most 7. In this paper, we verify the conjecture of Dvořák, Mohar and Šámal for a class of subcubic multigraphs. We prove that every claw-free subcubic multigraph has list star chromatic index at most 6, and give a few examples to show that the upper bound is tight. [ABSTRACT FROM AUTHOR]

We prove that the number of unit distances among n planar points is at most 1.94 • n4/3, improving on the previous best bound of 8 n4/3. We also give better upper and lower bounds for several small values of n. We also prove some variants of the crossing lemma and improve some constant factors. [ABSTRACT FROM AUTHOR]

Given a directed graph G and a pair of nodes s and t, an s-tbridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk W is safe with respect to a set W of s-t walks, if W is a subwalk of all walks in W . We start by considering the maximal safe walks when W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-tarticulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice. [ABSTRACT FROM AUTHOR]

RGB and thermal infrared (RGBT) image saliency detection is a relatively new direction in the field of computer vision. Combining the advantages of RGB images and T images can significantly improve detection performance. Currently, there are only a few methods to work on RGBT saliency detection, and the number of image samples cannot meet the training requirements for deep learning, so it remains valuable to propose an effective unsupervised method. In this paper, we present an unsupervised RGBT saliency detection method based on multi-graph fusion and learning. Firstly, RGB images and T images are adaptively fused based on boundary information to produce more accurate superpixels. Next, a multi-graph fusion model is proposed to selectively learn useful information from multi-modal images. Finally, we implement the theory of finding good neighbors in the graph affinity and propose different algorithms for two stages of saliency ranking. Experimental results on three RGBT datasets show that the proposed method is effective compared with the state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Recently, in Almallah et al. (New Math Nat Comput, to appear) and Borzooei et al. (New Math Nat Comput 16(2):397–418, 2020), we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory can not explain. And as we know planarity is very important and applicable concept in graph theory and fuzzy graph theory, so our motivation in this paper is discussion the importance of planarity in inverse fuzzy graphs. Now, in this paper we define the notion of inverse fuzzy multigraph and the concept of planarity on it by using the concepts of intersecting value and inverse fuzzy planarity value. Then we introduce some related theorems which determining upper bounds and lower bounds for the inverse fuzzy planarity value. After that we define the strong (weak) planarity of an inverse fuzzy multigraph and investigate related results. Finally, we give an application of inverse fuzzy multigraphs to decision-making how to reduce the cost of travel tours. [ABSTRACT FROM AUTHOR]

Aiming at the dynamic multicompartment refrigerated vehicle routing problem with multigraph based on real-time traffic information, this study, based on the idea of preoptimization followed by real-time adjustment, establishes a two-stage mathematical model with minimizing delivery cost. In the preoptimization phase, this study, based on historical traffic information, designed a hybrid chaotic genetic algorithm with variable neighborhood search (HCGAVNS) to obtain the initial delivery scheme. In the real-time adjustment phase, the order in which customers are served remains the same and a path selection strategy is proposed to solve the problem according to the real-time traffic information of different paths. The validity of the model and the algorithm are verified through the analysis of instances. The research results can enrich the related research on cold chain vehicle routing problem and provide a theoretical basis for logistics companies to optimize their delivery scheme. [ABSTRACT FROM AUTHOR]

Given a plane graph $G=(V,E)$ , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection ${\mathscr P}=\{P_1,\ldots,P_q\}$ of Petrie tours so that each edge of G is in exactly one tour $P_i \in {\mathscr P}$. A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection ${\mathscr C}=\{C_1,\ldots,C_p\}$ of Petrie cycles so that each vertex of G is in exactly one cycle $C_i \in {\mathscr C}$. In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph $G=(V,E)$ , a 3-regularization of G is a 3-regular plane graph $G_3$ obtained from G by splitting every vertex $v\in V$ into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip , used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete. [ABSTRACT FROM AUTHOR]

An energy hub, consisting of a combination of electricity, heat power, cooling power, natural gas, and other energy sources, is considered a key component of the energy internet. It requires quick and accurate optimization and control as well as a standardized and programmable model. This study proposes an intelligent modelling method based on a directed multigraph. This method starts from an input-output model and then establishes a directed multigraph in which a vertex indicates energy and an edge indicates energy conversion equipment and its parameters. Then, an adjacency matrix is obtained by processing and simplifying the directed multigraph. This adjacency matrix is searched using an intelligent algorithm to obtain the coupling matrix model of the energy hub. A hydrodynamic laboratory consisting of electricity, natural gas, heating, and cooling energy is used as a case study to verify the reliability and accuracy of the modelling process and to provide standardized data for deep learning uses in the energy internet. The obtained results show that the proposed method can quickly and effectively establish the energy hub model. This method is also effective when an energy storage device is added to or removed from the energy hub. [ABSTRACT FROM AUTHOR]

Band selection is an effective means to alleviate the curse of dimensionality in hyperspectral data. Many methods select a compact and low redundant band subset, which is inadequate as it may degrade the classification performance. Instead, more emphasis shall be put on selecting representative bands. In this article, we propose a robust unsupervised band selection method to address this issue. Our method reveals bandwise representativeness based on the comprehensive interband neighborhood structure. It incorporates an interband neighborhood graph into a sparse self-contained regression model in order to provide a reasonable measure for bandwise representativeness. The derived coefficient matrix not only uncovers bandwise importance values but also is coherent to the generalized interband local neighborhood structure. For constructing the interband neighboring structural graph, an integrated multigraph model is employed to achieve better generalization performance. It combines the benefit of multiple graphs but is insusceptible to the defects of a single one. To enhance the reliability of this model, a joint trace minimum and nonnegative constraint is imposed on the coefficient matrix. Accordingly, a multigraph integrated embedding and robust self-contained regression model (MGRSR) is formulated. In addition, an iterative update algorithm is developed to solve the problem. Comparative experiments on three hyperspectral data sets illustrate that MGRSR is robust to various data and has superior performance compared with several state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Data-driven transient stability assessment (TSA) models are usually sensitive to system scale changes and require dynamic information from time-domain simulation (TDS) as inputs. We propose a System-scAle-FreE Transient Contingency Screening (SAFE-TCS) scheme based on only the steady-state measurements. An analytical model is set up to estimate the state variation at fault occurrence ($t_{0+}$) snapshot, which forms multi-graph inputs together with the steady-state information. A novel pooling-ensemble multi-attention graph convolutional network (PE-MAGCN) realizes the spatio-temporal graph embedding, in which an inter-graph convolution link works for the temporal abstraction. Following a pooling-ensemble mechanism, PE-MAGCN derives a fixed-size expressive vector for task-specific networks. This promotes the robustness of the model against system extension. The advantages of SAFE-TCS also benefit from the coordination of various training tricks, including channel attention, category-balanced sampling and joint-decoupling learning, etc. Test results on IEEE 39 Bus system and IEEE 300 Bus system indicate the superiority of the proposed scheme over existing models and its adaptability under various scenarios. [ABSTRACT FROM AUTHOR]

Let Pk+1 denote a path of length k, Sm denote a star with m edges, and Kn(λ) denote the complete multigraph on n vertices in which every pair of distinct vertices is joined by λ edges. In this paper, we have obtained the necessary conditions for a {Pk+1, Sm}-decomposition of Kn(λ) and proved that the necessary conditions are also sufficient when k = 6 and m = 4. [ABSTRACT FROM AUTHOR]

Multivariate networks comprising several compositional and structural variables can be represented as multigraphs by various forms of aggregations based on vertex attributes. We propose a framework to perform exploratory and confirmatory multiplexity analysis of aggregated multigraphs in order to find relevant associations between vertex and edge attributes. The exploration is performed by comparing frequencies of the different edges within and between aggregated vertex categories, while the confirmatory analysis is performed using derived complexity or multiplexity statistics under different random multigraph models. These statistics are defined by the distribution of edge multiplicities and provide information on the covariation and dependencies of different edges given vertex attributes. The presented approach highlights the need to further analyse and model structural dependencies with respect to edge entrainment. We illustrate the approach by applying it on a well known multivariate network dataset which has previously been analysed in the context of multiplexity. [ABSTRACT FROM AUTHOR]

Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph G on n vertices whose contractions leave a multigraph with O ̃ (n / δ) vertices and O ̃ (n) edges that preserves all non-trivial min-cuts of G , where δ is the minimum degree of G and O ̃ hides logarithmic factors. We present a simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves O (n / δ) vertices and O (n) edges, preserves all non-trivial min-cuts and can be computed in near-linear time O ̃ (m) , where m is the number of edges of G. We also obtain that every simple graph has O ( (n / δ) 2) non-trivial min-cuts. Our approach allows to represent all non-trivial min-cuts of a graph by a cactus representation, whose cactus graph has O (n / δ) vertices. Moreover, this cactus representation can be derived directly from the standard cactus representation of all min-cuts in linear time. We apply this compact structure to show that all min-cuts can be explicitly listed in O ̃ (m) + O (n 2 / δ) time for every simple graph, which improves the previous best time bound O (n m) given by Gusfield and Naor. [ABSTRACT FROM AUTHOR]

In (J Graph Theory 4:241–242, 1980), Burr proved that χ (G) ≤ m 1 m 2 ... m k if and only if G is the edge-disjoint union of k graphs G 1 , G 2 , ... , G k such that χ (G i) ≤ m i for 1 ≤ i ≤ k . This result established the practice of describing the chromatic number of a graph G which is the edge-disjoint union of k subgraphs G 1 , G 2 , ... , G k in terms of the chromatic numbers of these subgraphs, and more specific results and conjectures followed. We investigate possible extensions of this theorem of Burr to group coloring and DP-coloring of multigraphs, as well as extensions of another vertex coloring theorem involving arboricity. In particular, we determine the DP-chromatic number of all Halin graphs. In (J Graph Theory 50:123–129, 2005), it is conjectured that for any graph G, the list chromatic number is not higher than the group chromatic number of G. As related results, we show that the group list chromatic number of all multigraphs is at most the DP-chromatic number, and present an example G for which the group chromatic number of G is less than the DP-chromatic number of G. [ABSTRACT FROM AUTHOR]

Let G be a graph that may have multiedges, let μ (u , v) denote the number of edges joining u and v in G, and let μ (u) = max v ∈ N (u) μ (u , v) . Let a , b : V (G) ⟶ N \ { 0 , 1 } be two functions with d G (v) ≥ a (v) + b (v) + 2 μ (v) - 3 for each v ∈ V (G) . We show that, if G does not contain C 4 , then G admits a partition (A, B) with d G [ A ] (u) ≥ a (u) for each u ∈ A , and d G [ B ] (v) ≥ b (v) for each v ∈ B . This generalizes a theorem of Ma and Yang on simple graphs, and answers a question of Schweser and Stiebitz. [ABSTRACT FROM AUTHOR]

We propose a novel method for large-scale image stitching that is robust against repetitive patterns and featureless regions in the imagery. In such cases, state-of-the-art image stitching methods easily produce image alignment artifacts, since they may produce false pairwise image registrations that are in conflict within the global connectivity graph. Our method augments the current methods by collecting all the plausible pairwise image registration candidates, among which globally consistent candidates are chosen. This enables the stitching process to determine the correct pairwise registrations by utilizing all the available information from the whole imagery, such as unambiguous registrations outside the repeating pattern and featureless regions. We formalize the method as a weighted multigraph whose nodes represent the individual image transformations from the composite image, and whose sets of multiple edges between two nodes represent all the plausible transformations between the pixel coordinates of the two images. The edge weights represent the plausibility of the transformations. The image transformations and the edge weights are solved from a non-linear minimization problem with linear constraints, for which a projection method is used. As an example, we apply the method in a large-scale scanning application where the transformations are primarily translations with only slight rotation and scaling component. Despite these simplifications, the state-of-the-art methods do not produce adequate results in such applications, since the image overlap is small, which can be featureless or repetitive, and misalignment artifacts and their concealment are unacceptable. [ABSTRACT FROM AUTHOR]

We prove #P-completeness results for counting edge colorings on simple graphs. These strengthen the corresponding results on multigraphs from [4]. We prove that for any κ ≥ r ≥ 3 counting κ -edge colorings on r -regular simple graphs is #P-complete. Furthermore, we show that for planar r -regular simple graphs, where r ∈ { 3 , 4 , 5 } , counting edge colorings with κ colors for any κ ≥ r is also #P-complete. As there are no planar r -regular simple graphs for any r > 5 , these statements cover all interesting cases in terms of the parameters (κ , r). [ABSTRACT FROM AUTHOR]

The Graph Convolutional Network (GCN) is a universal relation extraction method that can predict relations of entity pairs by capturing sentences' syntactic features. However, existing GCN methods often use dependency parsing to generate graph matrices and learn syntactic features. The quality of the dependency parsing will directly affect the accuracy of the graph matrix and change the whole GCN's performance. Because of the influence of noisy words and sentence length in the distant supervised dataset, using dependency parsing on sentences causes errors and leads to unreliable information. Therefore, it is difficult to obtain credible graph matrices and relational features for some special sentences. In this article, we present a Multi-Graph Cooperative Learning model (MGCL), which focuses on extracting the reliable syntactic features of relations by different graphs and harnessing them to improve the representations of sentences. We conduct experiments on a widely used real-world dataset, and the experimental results show that our model achieves the state-of-the-art performance of relation extraction. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of multiple graph matching (MGM) by considering both offline batch mode and online setting. We explore the concept of cycle-consistency over pairwise matchings and formulate the problem as finding optimal composition path on the supergraph, whose vertices refer to graphs and edge weights denote score function regarding consistency and affinity. By our theoretical study we show that the offline and online MGM on supergraph can be converted to finding all pairwise shortest paths and single-source shortest paths respectively. We adopt the Floyd algorithm and shortest path faster algorithm (SPFA) , to effectively find the optimal path. Extensive experimental results show our methods surpass state-of-the-art MGM methods, including CAO , MISM , IMGM , and many other recent methods in offline and online settings. Source code will be made publicly available. [ABSTRACT FROM AUTHOR]

With the advancement of Cyber Physic Systems and Social Internet of Things, the tourism industry is facing challenges and opportunities. We can now able to collect, store, and analyze large amounts of travel data. With the help of data science and artificial intelligence, smart tourism enables tourists with great autonomy and convenience for an intelligent trip. It is of great significance to make full use of these massive data to provide better services for smart tourism. However, due to the skewed and imbalanced visiting for point of interest located at different places, it is of great significance to predict the tourist flow of each place, which can help the service providers for designing a better schedule visiting strategy in advance. Against this background, this article proposes a multi-graph convolutional network framework, named AMOUNT, for tourist flow prediction. To capture the diverse relationships among POIs, AMOUNT first constructs three subgraphs, including the geographical graph, interaction graph, and the co-relation graph. Then, a multi-graph convolution network is utilized to predict the future tourist flow. Experimental results on two real-world datasets indicate that the proposed AMOUNT model outperforms all other baseline tourist flow prediction approaches. [ABSTRACT FROM AUTHOR]

In this paper, we studied a stochastic bi-objective mathematical model for effective and reliable rescue operations in multigraph network. The problem is addressed by a two-stage stochastic nonlinear mixed-integer program where the reliability of routes is explicitly traded-off with total weighted completion time. The underlying transportation network is able to keep a group of multiattribute parallel arcs between every pair of nodes. By this, the proposed model should consider the routing decision in logistic planning along with the path selection in an uncertain condition. The first stage of the model concerns with the vehicle routing decisions which is not involved with random parameters; besides, the second stage of the model involves with the departure time at each demand node and path finding decisions after observation of random vectors in the first stage considering a finite number of scenarios. To efficiently solve the presented model, an enhanced nondominated sorting genetic algorithm II (NSGA-II) is proposed. The effectiveness of the introduced method is then evaluated by conducting several numerical examples. The results implied the high performance of our method in comparison to the standard NSGA-II. In further analyses, we investigated the beneficiary of using multigraph setting and showed the applicability of the proposed model using a real transportation case. [ABSTRACT FROM AUTHOR]

We construct new resolvable decompositions of complete multigraphs and complete equipartite multi-graphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both com- plete multigraphs and complete equipartite multi- graphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions. [ABSTRACT FROM AUTHOR]

Product low carbon design plays a significant part in reducing carbon emissions during the product life cycle and achieving sustainable development. Aiming at the uncertainty of the carbon footprint caused by the unknown decision of product material solution, this paper proposes a low carbon design approach that uses the multigraph and colour marking to solve the problem of one edge assigned with multiple weights and the constraint problem of vertex combination in the decision space of product low carbon design. After a weighted edge-coloured graph (WECG) is proposed for low carbon design, the multi-stage product low carbon design model for product life cycle is discussed in detail. An algorithm of searching the alternative set of optimal design is also put forward in detail, including to create the directed WECG, to define the complete path, to find the ideal solution set using breadth-first search, to judge the ideal solution set by colouring the graph to obtain the feasible paths, and to rank paths in the ideal solution set in ascending order. The low carbon design of a robot is given as an example to demonstrate the methodology. [ABSTRACT FROM AUTHOR]

It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, every forest is pervasive, and K 3 is pervasive, in the class of all graphs, where we say a graph H is "pervasive" (in some class of graphs) if for all ℓ ≥ 1 , every graph in the class of bounded clique number and sufficiently large chromatic number has an induced subgraph that is a subdivision of H , in which every edge of H is replaced by a path of at least ℓ edges. Which other graphs are pervasive? It was proved by Chalopin, Esperet, Li and Ossona de Mendez that every such graph is a "forest of lanterns": roughly, every block is a "lantern", a graph obtained from a tree by adding one extra vertex, and there are rules about how the blocks fit together. It is not known whether every forest of lanterns is pervasive in the class of all graphs; but in another paper two of us prove that all "banana trees" are pervasive, that is, multigraphs obtained from a forest by adding parallel edges, thus generalizing the two results above. This paper contains the first half of the proof, which works for any forest of lanterns, not just for banana trees. Say a class of graphs is " ρ -controlled" if for every graph in the class, its chromatic number is at most some function (determined by the class) of the largest chromatic number of a ρ -ball in the graph. In this paper we prove that for every ρ ≥ 2 , and for every ρ -controlled class, every forest of lanterns is pervasive in this class. These results turn out particularly nicely when applied to string graphs. A "chandelier" is a special lantern, a graph obtained from a tree by adding a vertex adjacent to precisely the leaves of the tree. A "string graph" is the intersection graph of a set of curves in the plane. There are string graphs with clique number two and chromatic number arbitrarily large. We prove that the class of string graphs is 2-controlled, and consequently every forest of lanterns is pervasive in this class; but in fact something stronger is true, that every string graph of sufficiently large chromatic number and bounded clique number contains each fixed chandelier as an induced subgraph (not just as a subdivision); and the same for most forests of chandeliers (there is an extra condition on how the blocks are attached together). [ABSTRACT FROM AUTHOR]

Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\] , where \[c > 9/10\] , and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from Mi for each \[1 \leqslant i \leqslant m\]. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger. [ABSTRACT FROM AUTHOR]

• Multiple graphs are constructed with ridesplitting demand characteristics. • The spatial scope of the demand prediction is refined to the intersection level. • It proposes a MGCGRU-P model for the point and probabilistic demand prediction. • It measures the impact of graph combinations and correlation levels on predictions. • The efficiency of the proposed model is verified by a realistic case study. Ridesplitting demand prediction plays an important role in vehicle scheduling and intelligent transportation system construction. Accurate ridesplitting demand prediction is crucial for alleviating supply–demand imbalance and increasing vehicle utilization. Ridesplitting refers to the sharing of a vehicle by multiple passengers with similar routes, thereby achieving a reduction in the adverse effects of increased congestion caused by ride-hailing. However, most existing works mainly address the region-based ride-hailing demand prediction, whereas few works focus on the ridesplitting demand prediction, which takes into account the correlation between shared orders. To address the issue, we develop several intersection-based graphs to character the ridesplitting demand correlations between the intersections. The multi-graph convolutional networks (MGCN) with different graph combinations are constructed to capture the spatial topologies. Furthermore, we introduce the probabilistic model structure to predict the uncertainty of ridesplitting demand. A novel deep learning model, referred to as the Multi-graph Convolutional Gated Recurrent Unit with Probabilistic prediction (MGCGRU-P) is proposed. We conduct extensive numerical experiments on the real-world ride-hailing demand dataset (from Beijing, China). The results demonstrate that the model we propose performs the best against baseline models. [ABSTRACT FROM AUTHOR]