Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces

This article contributes to the study of graph rigidity and its interplay with fundamental graph invariants. Recently, a quantitative measure of graph rigidity in $R^d$, termed the generalized algebraic connectivity, was introduced. This development extends the notion of algebraic connectivity—the second-smallest eigenvalue of the Laplacian matrix. In this work, we show that the generalized algebraic connectivity is bounded above by the algebraic connectivity. To capture this relationship, we introduce the d-rigidity ratio, a normalized metric of a graph's rigidity relative to its connectivity. We also investigate the relationship between rigidity and the diameter. In this context, we provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Moreover, we establish a new upper bound for the algebraic connectivity that depends inversely on the diameter and the vertex connectivity. Finally, we derive an upper bound for the algebraic connectivity of generalized path graphs that asymptotically improves existing ones by a factor of four.