Let G be a simple graph of order n. The nullity of a graph G, denoted by $\eta (G)$, 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 $\mathrm{gr}(G)$, is the length of the shortest cycle in G. It is known that $\eta (G)$ is bounded above by $n-\mathrm{gr}(G)+2$ if $4|\mathrm{gr}(G)$ and by $n-\mathrm{gr}(G)$ if $4\nmid \mathrm{gr}(G)$. In this paper it is proved that when G is connected, $\eta (G)=n-\mathrm{gr}(G)+2$ 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 $\eta (G)\le n-\mathrm{gr}(G)$. Connected graphs of order n with girth g and nullity $n-g$ 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.

设G是一个简单的n阶图。图G的无效性，表示为$\eta (G)$, 是作为其邻接矩阵特征值的 0 的重数。如果G至少有一个圈，则G的周长，表示为$\mathrm{克}(G)$, 是G 中最短周期的长度。众所周知$\eta (G)$ 上界为 $n-\mathrm{克}(G)+2$ 如果 $4|\mathrm{克}(G)$ 并由 $n-\mathrm{克}(G)$ 如果 $4\nmid \mathrm{克}(G)$. 本文证明当G连通时，$\eta (G)=n-\mathrm{克}(G)+2$当且仅当G是一个完整的二部图，不同于一个星形，或者一个长度为 4 倍数的循环；如果G不是完全二部图或长度为 4 的倍数的环，则$\eta (G)\le n-\mathrm{克}(G)$. 具有周长g和零的n阶连通图$n-G$被表征。这项工作还解决了表征秩等于周长的连通图的问题，以及将包含给定非奇异环的所有连通图G识别为最短环且与G具有相同秩的问题。