Let G be a simple graph of order n. The nullity of a graph G, denoted by , is the multiplicity of 0 as an eigenvalue of its adjacency matrix. If G has at least one cycle, then the girth of G, denoted by , is the length of the shortest cycle in G. It is known that is bounded above by if and by if . In this paper it is proved that when G is connected, if and only if G is a complete bipartite graph, different from a star, or a cycle of length a multiple of 4; that if G is not a complete bipartite graph or a cycle of length a multiple of 4, then . Connected graphs of order n with girth g and nullity are characterized. This work also settles the problem of characterizing connected graphs with rank equal to girth and the problem of identifying all connected graphs G that contains a given nonsingular cycle as a shortest cycle and with the same rank as G.
