WebAlgebraic connectivity, the second smallest eigenvalue of the graph Laplacian matrix, is a fundamental performance measure in various network systems, such as multi-agent networked systems. Here, we focus on how to add an edge to a network to increase network connectivity and robustness by maximizing the algebraic connectivity. Most efficient …

Ordering trees and graphs with few cycles by algebraic connectivity ...

WebJun 1, 2015 · By introducing two notions of general algebraic connectivity, a detailed analysis has been performed to reach global synchronization. At the same time, the case of infinitely frequent triggering is excluded by showing that the inter-event interval is strictly larger than a positive low bound. It is found that some existing results can be seen ... WebIn this video, we look at how to compute the algebraic connectivity of a graph, which is equivalent to the second-smallest eigenvalue of the simple Laplacian... modulenotfounderror: no module named phonopy

Convex Optimization of Graph Laplacian Eigenvalues - Stanford …

WebJul 8, 2016 · The problem of connectivity assessment of an asymmetric network represented by a weighted directed graph is investigated in this paper. The notion of … In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 … See more Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti … See more For a non-negative integer k, the kth Betti number bk(X) of the space X is defined as the rank (number of linearly independent generators) of the abelian group Hk(X), the kth See more Betti numbers of a graph Consider a topological graph G in which the set of vertices is V, the set of edges is E, and the set of connected components is C. As explained in the … See more In geometric situations when $${\displaystyle X}$$ is a closed manifold, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms. … See more The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is $${\displaystyle 1+2x+x^{2}}$$. The same definition applies to any … See more 1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...; 2. The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... . See more • Topological data analysis • Torsion coefficient • Euler characteristic See more WebDec 1, 2024 · Here, Λ 2 is known as the algebraic connectivity of G. For a strongly connected directed graph, the general algebraic connectivity is defined as: a ... (34) κ > L o s a ξ (L) = κ ̄ 0, where a ξ (L) is the general algebraic connectivity of G … modulenotfounderror: no module named pyads