The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric matrices possess the Strong Spectral Property (denoted \({\mathcal {G}}^\textrm{SSP}\)) were studied, and along these lines we aim to study properties of graphs that exhibit a so-called barbell partition. Such a partition is a known impediment to membership in the class \({\mathcal {G}}^\textrm{SSP}\). In particular we consider the existence of barbell partitions under various standard and useful graph operations. We do so by considering both the preservation of an already present barbell partition after performing said graph operations as well as barbell partitions which are introduced under certain graph operations. The specific graph operations we consider are the addition and removal of vertices and edges, the duplication of vertices, as well as the Cartesian products, tensor products, strong products, corona products, joins, and vertex sums of two graphs. We also identify a correspondence between barbell partitions and graph substructures called forts, using this correspondence to further connect the study of zero forcing and the Strong Spectral Property.

满足强谱特性的矩阵的实用性已经得到很好的确立，特别是与图的逆特征值问题相关。最近，研究了所有相关对称矩阵都具有强谱特性（表示为\({\mathcal {G}}^\textrm{SSP}\) ）的图类，沿着这些思路，我们的目标是研究显示所谓杠铃分区的图表。这样的分区是类\({\mathcal {G}}^\textrm{SSP}\)中成员资格的已知障碍。特别是，我们考虑各种标准和有用的图形操作下杠铃分区的存在。我们通过考虑在执行所述图操作之后保留已经存在的杠铃分区以及在某些图操作下引入的杠铃分区来做到这一点。我们考虑的具体图操作是顶点和边的添加和删除、顶点的重复以及两个图的笛卡尔积、张量积、强积、电晕积、连接以及顶点和。我们还确定了杠铃分区和称为堡垒的图子结构之间的对应关系，利用这种对应关系进一步连接零强迫和强谱特性的研究。