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 (2012) 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.

中文翻译：

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刘易斯卡罗尔和红热土豆：线性代数恒等式的图论方法

摘要 刘易斯卡罗尔恒等式用通过删除一行和一列或一对行和列获得的子行列式来表达矩阵的行列式。使用矩阵树定理，我们可以将其转换为涉及森林对总和的等效恒等式。与刘易斯卡罗尔身份不同，森林身份不包含减号。2011 年，Vlasev 和 Yeats (2012) 建议可以使用类似于 Zeilberger 1997 年矩阵证明的边缘转移来证明这样的森林身份。但是，直到现在，还没有开发出这样的算法。在本文中，我们提供了这种边缘转移算法和 Lewis Carroll Identity 和 Forest Identity 的双射证明。这个双射是由红热土豆算法实现的，