  Over the past half century, the rigidity of graphs in $R^2$ has aroused a
great deal of interest. Lov\'{a}sz and Yemini (1982) proved that every
$6$-connected graph is rigid in $R^2$. Jackson and Jord\'{a}n (2005) provided a
similar vertex-connectivity condition for the globally rigidity of graphs in
$R^2$. These results imply that a graph $G$ with algebraic connectivity
$\mu(G)>5$ is (globally) rigid in $R^2$. Cioab\u{a}, Dewar and Gu (2021)
improved this bound, and proved that a graph $G$ with minimum degree
$\delta\geq 6$ is rigid in $R^2$ if $\mu(G)>2+\frac{1}{\delta-1}$, and is
globally rigid in $R^2$ if $\mu(G)>2+\frac{2}{\delta-1}$. In this paper, we
study the (globally) rigidity of graphs in $R^2$ from the viewpoint of
adjacency eigenvalues. Specifically, we provide sufficient conditions for a
2-connected (resp. 3-connected) graph with given minimum degree to be rigid
(resp. globally rigid) in terms of the spectral radius. Furthermore, we
determine the unique graph attaining the maximum spectral radius among all
minimally rigid graphs of order $n$.
