Algebraic connectivity in normed spaces

Abstract

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 $\ell_\infty^d$ 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 $\ell_\infty^d$ are odd-hole-free. Connections to redundant rigidity are also presented.