Pivots, Determinants, and Perfect Matchings of Graphs

TL;DR

This paper characterizes how pivot operations affect graphs using adjacency matrix determinants, linking pivots to perfect matchings and extending results to graphs with self-loops.

Contribution

It provides a novel characterization of pivot sequences via determinants and relates pivots to perfect matchings, including graphs with self-loops.

Findings

Two pivot sequences are equivalent iff they involve the same vertex set mod two.

Characterization of the existence of pivot sequences for a given vertex set.

Extension of results to graphs with self-loops and local complementations.

Abstract

We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations.