Spectral and Homological Bounds on k-Component Edge Connectivity (2409.05725v1)

Abstract: We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity to higher-dimensional simplicial complexes, we derive tight spectral-homological bounds on the minimum number of edges that must be removed to ensure that all remaining components in the graph have size less than k. These bounds relate the spectra of graph and simplicial Laplacians to topological invariants from homology, establishing a multi-dimensional measure of network robustness. Our framework improves the understanding of network resilience in critical systems such as the Western U.S. power grid and European rail network, and we extend our analysis to random graphs and expander graphs to demonstrate the broad applicability of the method. Keywords: k-component edge connectivity, spectral graph theory, homology, simplicial complexes, network resilience, Betti numbers, algebraic connectivity, random graphs, expander graphs, infrastructure systems