A New Upper Bound for the d-dimensional Algebraic Connectivity of Arbitrary Graphs

TL;DR

This paper establishes that the d-dimensional algebraic connectivity of any graph is always less than or equal to its 1-dimensional algebraic connectivity, providing a new upper bound based on eigenvalues of the graph Laplacian.

Contribution

It introduces a novel upper bound for the d-dimensional algebraic connectivity of arbitrary graphs, linking it to the well-known 1-dimensional case.

Findings

The d-dimensional algebraic connectivity is bounded above by the 1-dimensional algebraic connectivity.

This bound applies to arbitrary graphs, regardless of their structure.

The result simplifies understanding of higher-dimensional connectivity in graph theory.

Abstract

In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.