Note on the rigidity of graphs

Abstract

It is of interest to look for the sufficient conditions for the rigidity of a graph. Fan, Huang and Lin (2023) recently studied the rigidity of a graph from the perspective of its spectral radius of the adjacency matrix and established a sufficient condition involving the spectral radius to ensure a $2$-connected (or a $3$-connected) graph $G$ with a fixed minimum degree to be rigid (or globally rigid). In this note, we establish a similar condition which relates $\lambda_1^\alpha(G)$, the spectral radius of the matrix $A_\alpha(G):= \alpha D(G) + (1
-\alpha)A(G)$, where $\alpha \in (0, 1)$, $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal degree matrix of $G$,
respectively.