On the $d$-dimensional algebraic connectivity of graphs

TL;DR

This paper investigates the $d$-dimensional algebraic connectivity of complete graphs, providing exact values for certain cases and bounds for general $n$, which relate to the graph's rigidity in $d$-dimensional space.

Contribution

It offers new bounds and exact values for the $d$-dimensional algebraic connectivity of complete graphs, advancing understanding of their rigidity properties.

Findings

For $d geq 3$, $a_d(K_{d+1})=1$.

For $n geq 2d$, bounds on $a_d(K_n)$ are established.

Provides explicit bounds relating $a_d(K_n)$ to $n$ and $d$.

Abstract

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$, introduced by Jord\'an and Tanigawa, is a quantitative measure of the $d$-dimensional rigidity of $G$ that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set $V$ into $\mathbb{R}^d$. Here, we analyze the $d$-dimensional algebraic connectivity of complete graphs. In particular, we show that, for $d\geq 3$, $a_d(K_{d+1})=1$, and for $n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq \frac{2n}{3(d-1)}+\frac{1}{3}. \]