Lewis Carroll and the Red Hot Potato: a graph theoretic approach to a linear algebraic identity

TL;DR

This paper introduces a novel graph-theoretic algorithm called the Red Hot Potato to provide bijective proofs of the Lewis Carroll and Forest Identities, connecting linear algebra and combinatorics.

Contribution

It develops the first edge transfer algorithm and bijective proof for these identities, bridging matrix theory and graph theory through the Red Hot Potato method.

Findings

Established a bijective proof for the Lewis Carroll Identity.

Created the Red Hot Potato algorithm for edge transfers.

Connected linear algebra identities with combinatorial graph structures.

Abstract

The Lewis Carroll Identity expresses the determinant of a matrix in terms of subdeterminants obtained by deleting one row and column or a pair of rows and columns. Using the matrix tree theorem, we can convert this into an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll Identity, the Forest Identity involves no minus signs. In 2011, Vlasev and Yeats suggested that such a Forest Identity could be proven using edge transfers similar to Zeilberger's 1997 matrix proof. However, until now, such an algorithm has not yet been developed. In this paper, we provide this edge transfer algorithm and a bijective proof for both the Lewis Carroll Identity and Forest Identity. This bijection is implemented by the Red Hot Potato algorithm, so called because the way edges get tossed back and forth between the two forests is reminiscent of the children's game of hot potato.