Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The positive semidefiniteness of Laplacian matrices corresponding to graphs with negative edge ights is studied. Two alternative proofs to a result by Zelazo and Burger (Theorem 3.2), which provides upper bounds on the magnitudes of the negative ights in terms of effective resistances within which to ensure definiteness of the Laplacians, are provided. Both proofs are direct and intuitive. The first employs purely geometrical arguments while the second relies on passivity arguments and the laws of physics for electrical circuits. The latter is then used to establish consensus in multi-agent systems with generalized high-order dynamics. A numerical example is given at the end of the paper to highlight the result.
['Yongxin Chen', 'Sei Zhen Khong', 'Tryphon T. Georgiou']
I am writing a research paper on a idea On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches
[{"from": "human", "value": "I am writing a research paper on a idea On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The positive semidefiniteness of Laplacian matrices corresponding to graphs with negative edge ights is studied. Two alternative proofs to a result by Zelazo and Burger (Theorem 3.2), which provides upper bounds on the magnitudes of the negative ights in terms of effective resistances within which to ensure definiteness of the Laplacians, are provided. Both proofs are direct and intuitive. The first employs purely geometrical arguments while the second relies on passivity arguments and the laws of physics for electrical circuits. The latter is then used to establish consensus in multi-agent systems with generalized high-order dynamics. A numerical example is given at the end of the paper to highlight the result."}]