Paths are generically realisable

TL;DR

This paper proves that any 0-1 multiplicity matrix for a simple graph, including paths, can be realized through a generic matrix, and applies this to identify graph families with matrices having only two eigenvalues.

Contribution

It establishes the generic realizability of all 0-1 multiplicity matrices for simple graphs, including paths, and explores applications to graphs with matrices having two eigenvalues.

Findings

All 0-1 multiplicity matrices for simple graphs are generically realizable.

Paths have multiplicity matrices that are generically realizable.

Certain graph families can be realized with matrices having only two distinct eigenvalues.

Abstract

We show that every $0$-$1$ multiplicity matrix for a simple graph $G$ is generically realisable for $G$. In particular, every multiplicity matrix for a path is generically realisable. We use this result to provide several families of joins of graphs that are realisable by a matrix with only two distinct eigenvalues.