The Alternative of Sensor Placement in Multi-Story Buildings through the Metric Dimension Approach: A Representation of Generalized Petersen Graphs

In a public facility or private office where many people can get together, a fire detection device is a mandatory tool as an emergency alarm in the facility. However, the expense of the installation of the device is a troublesome matter. So, optimization is needed to minimize the number of these devices. The way to implement is to select the appropriate position to place the devices in public facilities. The research discussed the placement of the sensors in multi-story buildings. The multi-story buildings could be represented as cube composition graphs with the number of rooms, and the connectivity between the floor and its rooms was equal. The concept of this multi-story building was modeled into a generalized Petersen graph where a vertex represented a room, and an edge was the connectivity of rooms. The basis obtained on that metric dimension was represented as a sensor placed on the building. Then, the optimization of device placement was seen as determining the metric dimensions of the Petersen graph. In the research, the alternative sensor placements were computed using the graph metric dimension approach implemented in Python. The research successfully implements the metric dimension of  to  using Python code to obtain the alternative of its basis. A basic alternative indicates the location of the device placement like fire detectors, network access points, or other sensors inside a building

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Notes on the Localization of Generalized Hexagonal Cellular Networks

The act of accessing the exact location, or position, of a node in a network is known as the localization of a network. In this methodology, the precise location of each node within a network can be made in the terms of certain chosen nodes in a subset. This subset is known as the locating set and its minimum cardinality is called the locating number of a network. The generalized hexagonal cellular network is a novel structure for the planning and analysis of a network. In this work, we considered conducting the localization of a generalized hexagonal cellular network. Moreover, we determined and proved the exact locating number for this network. Furthermore, in this technique, each node of a generalized hexagonal cellular network can be accessed uniquely. Lastly, we also discussed the generalized version of the locating set and locating number

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Graph Deep Learning: State of the Art and Challenges

The last half-decade has seen a surge in deep learning research on irregular domains and efforts to extend convolutional neural networks (CNNs) to work on irregularly structured data. The graph has emerged as a particularly useful geometrical object in deep learning, able to represent a variety of irregular domains well. Graphs can represent various complex systems, from molecular structure, to computer and social and traffic networks. Consequent on the extension of CNNs to graphs, a great amount of research has been published that improves the inferential power and computational efficiency of graph- based convolutional neural networks (GCNNs).The research is incipient, however, and our understanding is relatively rudimentary. The majority of GCNNs are designed to operate with certain properties. In this survey we review of the state of graph representation learning from the perspective of deep learning. We consider challenges in graph deep learning that have been neglected in the majority of work, largely because of the numerous theoretical difficulties they present. We identify four major challenges in graph deep learning: dynamic and evolving graphs, learning with edge signals and information, graph estimation, and the generalization of graph models. For each problem we discuss the theoretical and practical issues, survey the relevant research, while highlighting the limitations of the state of the art. Advances on these challenges would permit GCNNs to be applied to wider range of domains, in situations where graph models have previously been limited owing to the obstructions to applying a model owing to the domains’ natures