Let \(\mathscr {B}\) denote the class of bicyclic graphs with a unique perfect matching. A matching edge [i, j] is called a peg of a cycle if exactly one of i or j lies on the cycle or a chord. We show that the diagonal entries of the inverse of the adjacency matrix of a bicyclic graph \(B \in \mathscr {B}\) are zero whenever every cycle in B contains at least two pegs. We denote this class by \(\mathscr {B}_0\) . Within this class, we characterize the graphs that admit bicyclic inverses. In particular, a graph in \(\mathscr {B}_0\) without a 4-cycle has a bicyclic inverse if and only if it is a simple corona. We also characterize the non-corona bicyclic graphs in this class that possess bicyclic inverses and provide all possible constructions of such graphs.
