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Graph-Theoretic Algorithms for Energy Saving in IP Networks
Published in F. Richard Yu, Xi Zhang, Victor C. M. Leung, Green Communications and Networking, 2016
Francesca Cuomo, Antonio Cianfrani, Marco Polverini
where (n, m) indicates the link between node n and m. The adjacency matrix of a simple bidirectional graph is symmetric and has all diagonal elements equal to 0. The degree matrix D(G) of G is a diagonal matrix with the generic element dnn equal to the degree of node n, i.e., to the number of edges incident on n: dnn=Σm=1Nanm=Σm=1Namn.
Partitioning and Clustering
Published in Charles J. Alpert, Dinesh P. Mehta, Sachin S. Sapatnekar, Handbook of Algorithms for Physical Design Automation, 2008
We denote the ith eigenvalue of A by αiDefinition 8Given an adjacency matrix of a graph on |V| vertices, the diagonal degree matrix, D, is defined asdii=∑j=1|V|aijDefinition 9Given a graph on |V| vertices, define the corresponding |V| × |V| Laplacian matrix, L = D − A.
Multi-Aerial-Robot Planning
Published in Yasmina Bestaoui Sebbane, Multi-UAV Planning and Task Allocation, 2020
where L = D − A is the graph Laplacian matrix, D = diag(A) is the degree matrix with diagonal elements di = ∑jaij and 1 = (1,..., 1) is the vector of ones. By definition, L1 and thus graph Laplacian always have a zero eigenvalue λ1 = 0. If G is a strongly connected graph or has a directed spanning tree, rank(K) = n − 1 and all other eigenvalues of L are non-zero [83]. An equilibrium of system equation (1.20) is in a state X* = (ζ ,...,ζ) = ζ1 = (1,..., 1) where all nodes agree.
Pattern formation of multi-AUV systems with the optical sensor based on displacement-based formation control
Published in International Journal of Systems Science, 2020
Xiaomin Wang, Benoit Zerr, Héléne Thomas, Benoit Clement, Zexiao Xie
The Laplacian matrix of a graph is defined as , where is the degree matrix of with the diagonal elements .
Consensus tracking for multi-agent systems subject to channel fading: a sliding mode control method
Published in International Journal of Systems Science, 2020
Xiaowei Gu, Tinggang Jia, Yugang Niu
The graph can be divided into undirected graph and directed graph. A weighted directed graph G consists of a pair , where and , respectively, represent the nonempty set of nodes and the set of edges. A directed edge in the graph G means that agent i can obtain the information from agent j, but not conversely. In the directed graph G, if there exists a directed path from i to j, there is a sequence of successive edges from i to j. The directed graph is called as strongly connected if there exists a directed path between each pair of distinct agents. In addition, the directed graph has a directed spanning tree if there exists at least one root node that has a directed path to every other node. The adjacency matrix of the graph G is defined by , where if and if or i = j. The degree matrix is defined as , where is the in-degree of node i. The Laplacian matrix of the graph G is designed as , where and , , . Moreover, the Laplacian matrix is also denoted as L = D−A. According to the definition of the Laplacian matrix, we have , where and have appropriate dimensions and ‘·’ represents the product of the matrix L and the vector .