In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new. the discovery of new formulas and we indeed provide at least a few new identities, which although not definitive, show some insight into the combinatorics of graphs with parameters and the power of Zeilberger's approach. The note is organised as follows. We first recall definitions about determinants, then proceed to give

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... r revisited and unified proof. Last, we prove new results and discuss in an open-ended way some natural perspectives.