We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7－ε)|P| and (11/6－ε)|P| are lower bounds for the maximum total number of edge crossings for any ε ＞ 0 in cases (k,ℓ)=(2,3) and (2,2), respectively. We also show that the lower bound for the maximum geometric thickness is 3 for both cases. In the proofs, we apply the method of arranging isomorphic units regularly. While the method is developed for the proof in case (k,ℓ)=(2,3), it also works for different ℓ.

In recent years, deep learning based approaches have substantially improved the performance of face recognition. Most existing deep learning techniques work well, but neglect effective utilization of face correlation information. The resulting performance loss is noteworthy for personal appearance variations caused by factors such as illumination, pose, occlusion, and misalignment. We believe that face correlation information should be introduced to solve this network performance problem originating from by intra-personal variations. Recently, graph deep learning approaches have emerged for representing structured graph data. A graph is a powerful tool for representing complex information of the face image. In this paper, we survey the recent research related to the graph structure of Convolutional Neural Networks and try to devise a definition of graph structure included in Compressed Sensing and Deep Learning. This paper devoted to the story explain of two properties of our graph - sparse and depth. Sparse can be advantageous since features are more likely to be linearly separable and they are more robust. The depth means that this is a multi-resolution multi-channel learning process. We think that sparse graph based deep neural network can more effectively make similar objects to attract each other, the relative, different objects mutually exclusive, similar to a better sparse multi-resolution clustering. Based on this concept, we propose a sparse graph representation based on the face correlation information that is embedded via the sparse reconstruction and deep learning within an irregular domain. The resulting classification is remarkably robust. The proposed method achieves high recognition rates of 99.61% (94.67%) on the benchmark LFW (YTF) facial evaluation database.