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 λ_1^a(G) , the spectral radius of the matrix A_a(G) := aD(G) + (1 - α)A(G) , where α ∈ (0, 1) , A(G) and D(G) are the adjacency matrix and the diagonal degree matrix of G, respectively.