Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices

Let G be a undirected graph without loops and multiple edges. By η(G),θ(G) and p(G) we respectively denote the nullity, the dimension of cycle space, and the number of pendant vertices of G. If each component of G contains at least two vertices, then it is proved that η(G)≤2θ(G)+p(G), the equality is attained if and only if every component of G is a cycle with size a multiple of 4.