The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions

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.