A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14. In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft

Classification of Geometric Forms in Mosaics Using Deep Neural Network

The paper addresses an image processing problem in the field of fine arts. In particular, a deep learning-based technique to classify geometric forms of artworks, such as paintings and mosaics, is presented. We proposed and tested a convolutional neural network (CNN)-based framework that autonomously quantifies the feature map and classifies it. Convolution, pooling and dense layers are three distinct categories of levels that generate attributes from the dataset images by introducing certain specified filters. As a case study, a Roman mosaic is considered, which is digitally reconstructed by close-range photogrammetry based on standard photos. During the digital transformation from a 2D perspective view of the mosaic into an orthophoto, each photo is rectified (i.e., it is an orthogonal projection of the real photo on the plane of the mosaic). Image samples of the geometric forms, e.g., triangles, squares, circles, octagons and leaves, even if they are partially deformed, were extracted from both the original and the rectified photos and originated the dataset for testing the CNN-based approach. The proposed method has proved to be robust enough to analyze the mosaic geometric forms, with an accuracy higher than 97%. Furthermore, the performance of the proposed method was compared with standard deep learning frameworks. Due to the promising results, this method can be applied to many other pattern identification problems related to artworks

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14. In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer Reviewe