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

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., $a_d(G)\le a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.