Bilangan Kromatik Pewarnaan Titik pada Graf Dual dari Graf Roda

Penelitian ini bertujuan mengkonstruksi graf dual dari graf roda (Wn*) dan menentukan bilangan kromatik graf dual dari graf roda (Wn*). Penelitian ini dimulai dari menggambarkan beberapa graf roda dari ke , kemudian membangun graf dual dari graf roda dengan memanfaatkan graf-graf dari ke , kemudian memberikan warna pada titik-titik dari graf dualnya dengan menentukan bilangan kromatiknya. Diperoleh hasil bahwa Graf roda merupakan graf self-dual karena isomorfik dengan graf dualnya yaitu . Pewarnaan titik diperoleh dengan menentukan bilangan kromatik graf dual dari graf roda, menentukan pola dari bilangan kromatik, dan memberikan warna. Berdasarkan hasil penelitian, diperoleh bilangan kromatik pewarnaan titik pada graf dual dari graf roda yakni Kata Kunci: Pewarnaan Titik, Bilangan Kromatik, Graf Dual dan Graf Roda.This research aims to construct a dual graph from a wheel graph (Wn*) and determine the dual graph chromatic number of the wheel graph (Wn*). This research starts from describing some wheel graph from to , then construct a dual graph from a wheel graph from to , then gives color to the vertices of the dual graph by determining the chromatic number. The result showed that the wheel graph is a self-dual graph because it is isomorphic with its dual graph, namely . The vertex coloring is obtained by determining the chromatic number of the dual graph of the wheel graph, determining the pattern of the chromatic number and giving the color. Based on the research results, the chromatic number of vertex coloring on dual graph of a wheel graph is: Keywords: Vertex Coloring, Chromatic Number, Dual Graph and Wheel Graph.

Dual graph wavelet neural network for graph-based semi-supervised classification

Vertex classification is an important graph mining technique and has important applications in fields such as social recommendation and e-Commerce recommendation. Existing classification methods fail to make full use of the graph topology to improve the classification performance. To alleviate it, we propose a Dual Graph Wavelet neural Network composed of two identical graph wavelet neural networks sharing network parameters. These two networks are integrated with a semi-supervised loss function and carry out supervised learning and unsupervised learning on two matrixes representing the graph topology extracted from the same graph dataset, respectively. One matrix embeds the local consistency information and the other the global consistency information. To reduce the computational complexity of the convolution operation of the graph wavelet neural network, we design an approximate scheme based on the first type Chebyshev polynomial. Experimental results show that the proposed network significantly outperforms the state-of-the-art approaches for vertex classification on all three benchmark datasets and the proposed approximation scheme is validated for datasets with low vertex average degree when the approximation order is small.

Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.

BUILDING ROOF VECTORIZATION WITH PPGNET

A challenge in data-based 3D building reconstruction is to find the exact edges of roof facet polygons. Although these edges are visible in orthoimages, convolution-based edge detectors also find many other edges due to shadows and textures. In this feasibility study, we apply machine learning to solve this problem. Recently, neural networks have been introduced that not only detect edges in images, but also assemble the edges into a graph. When applied to roof reconstruction, the vertices of the dual graph represent the roof facets. In this study, we apply the Point-Pair Graph Network (PPGNet) to orthoimages of buildings and evaluate the quality of the detected edge graphs. The initial results are promising, and adjusting the training parameters further improved the results. However, in some cases, additional work, such as post-processing, is required to reliably find all vertices.