The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $X\times (V\left(G\right)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is 1 if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V\left(G\right)\setminus X$ and 0 otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $\alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $\alpha$ for each real $\alpha \ge 0$. Finally, we describe explicitly all graphs of average cut-rank at most $3∕2$ and determine up to $3∕2$ all possible values that can be realized as the average cut-rank of some graph.

一组的最高级 $X$ 图中的顶点数 $G$ 被定义为 $X\times （V（G）\setminus X）$ 二进制字段上的矩阵 $（\mathrm{一世}，Ĵ）$-entry为1（如果顶点） $\mathrm{一世}$ 在 $X$ 与顶点相邻 $Ĵ$ 在 $V（G）\setminus X$否则为0。我们引入了称为图的平均割等级的图形参数，定义为随机顶点集的割等级的期望值。我们表明，当采用图的顶点小点时，并且仅当它具有有限的邻域多样性时，该参数的类别才具有有限的平均割秩，因此该参数不会增加。这使我们可以推断出每个实数$\alpha$，其平均割秩大于（或至少）的诱导子图最小图的列表 $\alpha$是有限的。我们通过提供最多（或小于）平均等级的障碍物大小的上限和障碍物数量的下限来进一步完善此功能$\alpha$ 对于每个真实 $\alpha \ge 0$。最后，我们明确描述最多所有平均排名的图表$3∕2$ 并确定 $3∕2$ 所有可能的值，可以将其实现为某张图的平均切割等级。