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

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