TY - JOUR
T1 - The Strong Spectral Property of Graphs
T2 - Graph Operations and Barbell Partitions
AU - Allred, Sarah
AU - Curl, Emelie
AU - Fallat, Shaun
AU - Nasserasr, Shahla
AU - Schuerger, Houston
AU - Villagrán, Ralihe R.
AU - Vishwakarma, Prateek K.
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2024.
PY - 2024/4
Y1 - 2024/4
N2 - 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 GSSP) 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 GSSP. 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.
AB - 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 GSSP) 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 GSSP. 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.
KW - Barbell partitions
KW - Eigenvalues
KW - Graph operations
KW - Graphs
KW - Strong Spectral Property
UR - http://www.scopus.com/inward/record.url?scp=85187129680&partnerID=8YFLogxK
U2 - 10.1007/s00373-023-02745-6
DO - 10.1007/s00373-023-02745-6
M3 - Article
AN - SCOPUS:85187129680
SN - 0911-0119
VL - 40
JO - Graphs and Combinatorics
JF - Graphs and Combinatorics
IS - 2
M1 - 20
ER -