  The Hoffman ratio bound, Lov\'{a}sz theta function and Schrijver theta
function are classical upper bounds for the independence number of graphs,
which are useful in graph theory, extremal combinatorics and information
theory. By using generalized inverses and eigenvalues of graph matrices, we
give bounds for independence sets and the independence number of graphs. Our
bounds unify the Lov\'{a}sz theta function, Schrijver theta function and
Hoffman-type bounds, and we obtain the necessary and sufficient conditions of
graphs attaining these bounds. Our work leads to some simple structural and
spectral conditions for determining a maximum independent set, the independence
number, the Shannon capacity and the Lov\'{a}sz theta function of a graph.
