There is a conjecture on the signature:-c₃(G) ≤ s(G) ≤ c₅(G).Based on the eigenvalue interleaving theorem and the relationship between the rank and the signature,the induction method was used to prove that if there exists a vertex v in G satisfying d(v) < n-1 and r(G)≠r(G-v)+1,then the conjecture holds.Furthemore,some examples were given to demonstrate the existence of the mentioned graphs.Meanwhile,it was proved that if H is a k-cyclic graph with base χ_H,if there exists a vertex v on χ_H such that v is matched in H{v},then the graph H also coincides with the conjecture.