A new proof of Bartholdi's theorem (Q2574980)

Bartholdi's theorem gives two formulas for the reciprocal of the Bartholdi zeta function of a connected graph \(G\), in terms of the order and size of \(G\) and various matrices [\textit{L. Bartholdi}, Enseign. Math., II. Sér. 45, No. 1--2, 83--131 (1999; Zbl 0961.05032)]. The present authors give a new and shorter proof of this theorem, using linear algebra for the first formula, and a combinatorial argument for the second one.