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

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.