Rigid linkages and partial zero forcing

TL;DR

This paper introduces the concept of rigid linkages in graphs and connects them to zero forcing, providing new bounds on eigenvalue multiplicities of matrices associated with the graph.

Contribution

It defines rigid linkages as a new class of linkages, relates them to zero forcing, and uses them to derive bounds on eigenvalue multiplicities.

Findings

Rigid linkages are a special kind of unique linkage.

Spanning forcing paths form a spanning rigid linkage.

Rigid linkages help bound eigenvalue multiplicities.

Abstract

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph. Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.