2025/07/31 by James Cruickshank, Cruickshank, James, Sean Dewar +3
Engineering · Mathematics · #05C22 #05C50 #46B20 #52C25 #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Spectral Theory (math.SP) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2508.00134
openalex publication_date 2025/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The algebraic connectivity of a graph G in a finite dimensional real normed linear space X is a geometric counterpart to the Fiedler number of the graph and can be regarded as a measure of the rigidity of the graph in X. We analyse the behaviour of the algebraic connectivity of G in X with respect to graph decomposition, vertex deletion and isometric isomorphism, and provide a general bound expressed in terms of the geometry of X and the Fiedler number of the graph. Particular focus is given to the space ℓ_∞d where we present explicit formulae and calculations as well as upper and lower bounds. As a key tool, we show that the monochrome subgraphs of a complete framework in ℓ_∞d are odd-hole-free. Connections to redundant rigidity are also presented.