New proofs of Bapat and Sivasubramanian's theorems

Recently, Bapat and Sivasubramanian [1] presented formulas for the determinant and the inverse of the product distance matrix of a tree. Furthermore, they defined a bivariant Ihara–Selberg zeta function of a graph and gave its determinant expression. We present new proofs for three results of Bapat and Sivasubramanian.