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Role of graphs for multi-agent systems and generalization of Euler's Formula

机译：图在多智能体系统中的作用和欧拉公式的推广

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摘要

The historical problem of Seven Bridges of Königsberg caused the birth of graph theory. A number of practical problems involving networks may be appropriately represented by the graphs that facilitate problem formulation and analysis process. Communication topology for networks involving a large number of units, like multi-agent system and swarm of aerial vehicles etc., may be conveniently examined using the notion of graph theory. To facilitate the formulation of such problems, an appropriate mathematical solution is to represent the graph with the help of Laplacian matrix. Eigenvalues of Laplacian matrix are the main focus of this paper. Same have been exploited to give an insight into the graph / subgraph properties, and to generalize the well-known Euler's Formula in order to make it applicable for graphs as well as subgraphs. A modified Euler's formula is also presented. Effects of addition and removal of communication links for a given number of agents are examined. Effects of addition and removal of agents, and portioning a graph into subgraphs is also the focus of our present study.
机译：柯尼斯堡七桥的历史问题引起了图论的诞生。涉及网络的许多实际问题可以由有助于问题表述和分析过程的图表适当地表示。可以使用图论的概念方便地检查涉及大量单元的网络的通信拓扑，例如多智能体系统和飞行器群等。为了促进此类问题的形成，适当的数学解决方案是借助拉普拉斯矩阵表示图。拉普拉斯矩阵的特征值是本文的重点。利用相同的方法可以深入了解图形/子图形的属性，并归纳众所周知的欧拉公式，以使其适用于图形和子图形。还提出了修改后的欧拉公式。对于给定数量的代理，将检查添加和删除通信链接的效果。添加和删​​除代理以及将图形分为子图的效果也是我们当前研究的重点。

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