Note on the rigidity of graphs

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 ₁ᵃ (G), the spectral radius of the matrix Aₐ (G): = aD (G) + (1 -) A (G), where (0, 1), A (G) and D (G) are the adjacency matrix and the diagonal degree matrix of G, respectively.