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Topological extensions of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
In knot theory, Thistlethwaite's theorem relates the Jones polynomial and the Tutte polynomial of a corresponding planar graph as in Chapter 18. I. Pak suggested using the Bollobás–Riordan polynomial for Thistlethwaite-type theorems. This idea was first realized in [309] for a special class of (checkerboard colorable) virtual links. Then it was realized for classical links in [360], and for arbitrary virtual links in [310]. Formally all three theorems from [309, 310, 360] were different. They used different constructions of a ribbon graph from a link diagram and different substitutions in the Bollobás–Riordan polynomials of these graphs. An attempt to understand and unify these theorems led to the discovery of partial duality in [306] (called generalized duality there) and to a proof of an invariance of a certain specializations of the Bollobás–Riordan polynomial under it. Partial duality, which arose from graph polynomials, has proven to be an important construction in topological graph theory.
Application of graph databases in the communication and information asset management in power grid
Published in Lin Liu, Automotive, Mechanical and Electrical Engineering, 2017
Xuming Lv, Shanqi Zheng, Zhao Li, Siyan Liu, Yue Wang
To the best of knowledge, graph databases have not been widely applied in power system yet. However, communication and information in power system can be modelled as property graph. Devices, e.g., server, storage device, and network switch can be modelled as node. The transmission line and the logistical connection between devices can be modelled as edge. The configuration of the device, the power grid management system it belongs to, the actual address can be modelled as properties. To the best of our knowledge, we firstly utilized graph databases to study the communication and information asset management in power system. Firstly, we investigate the actual requirements of communication and information asset management in power system and conduct modelling. Secondly, we make a brief survey of the latest popular graph databases, i.e., Neo4j, GraphFrame, GraphSQL. Thirdly, we realize the modelling of the topological graph of the communication and information asset of power system. Fourthly, we achieve the visualization of communication and information asset managements
Data Fields and Topological Potential
Published in Deyi Li, Yi Du, Artificial Intelligence with Uncertainty, 2017
Through modeling the entry link relationship network in Chinese Wikipedia for the years 2003–2009, we can use the method for ranking node importance based on topological potential, as proposed in Section 4.3.2, to mine the hot entries and reveal development trends in computer science in different years. Figure 4.35 shows the topological graph for popular Wikipedia entries from 2003 to 2009. Because of large number of nodes, and to mark the changing trend in hot entries in the network, the diameter of each node in the graph is positively proportional to its topological potential value; the greater the topological potential value, the greater the diameter of the corresponding node, indicating it has a more important position in the network.
Unmanned Aerial Vehicle (UAV) path planning and control assisted by Augmented Reality (AR): the case of indoor drones
Published in International Journal of Production Research, 2023
Dimitris Mourtzis, John Angelopoulos, Nikos Panopoulos
The geometric modelling technique describes obstacles in a 3D world using geometric shapes (points, lines, and surfaces) (Yadgarov U.T. 2022). Geometric components and their connections are used to depict the internal information of a building. Complex obstacle bounds, like erratic curving surfaces, can cause modelling deviation. The topological graph technique uses graph theory to abstract the entire airspace into a graph (Zhang, Liu, and Tang 2018). Path planning development becomes extremely difficult in complicated airspace environments, particularly in 3D indoor spaces. The grid technique divides the entire airspace into uniform 2D or 3D grids, offering an efficient data model for pre-flight UAV conflict detection and path planning activities (Wu et al. 2022). The development of a path planning algorithm is the primary goal of path search, which is the central aspect of path planning (Yonetani et al. 2021). Path planning in narrow indoor environments must ensure that every step of a UAV flight is accurate and safe in the 3D flying airspace. In general, path planning algorithms necessitate the creation of a cost function or reward space (Zhou et al. 2020). The cost functions reflect the numerous dangers that a UAV faces as well as the many limits on the flying path. In general, the lower the cost function, the better the track and the level of implementation (Zhong et al. 2020). Summarising, three types of indoor planning algorithms are used in literature: Bio intelligence algorithms (Miao et al. 2021; Pan, Yang, and Li 2021)Neural networks or reinforcement learning algorithms (Arulkumaran et al. 2017; Horn et al. 2012)Searching algorithms (Shin and Kim 2021; Xiao, Tan, and Wang 2021)