Algebraic connectivity in normed spaces

TL;DR

This paper explores the algebraic connectivity of graphs within finite-dimensional normed spaces, analyzing its properties, bounds, and specific cases like _\u221e^d, linking geometric and graph rigidity concepts.

Contribution

It introduces a general framework for understanding algebraic connectivity in normed spaces, providing bounds, explicit formulas in _\u221e^d, and connections to rigidity theory.

Findings

Derived bounds based on space geometry and Fiedler number

Explicit formulas and bounds for _^d

Monochrome subgraphs are odd-hole-free in _^d

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.