Stability mapping of bipartite tight-binding graphs with losses and gain: $P T P T$-symmetry and beyond

We consider bipartite tight-binding graphs composed by $N$ nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability $p$⁠. Specifically, we measure the connectivity between the two sets with the parameter $α$⁠, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-Hermitian Hamiltonian $H(γ,α,N)$ of this model presents pseudo-Hermiticity, where $γ$ is the loss/gain strength. However, we show that for a given graph setup $H(γ,α,N)$ becomes $P T$-symmetric. In both scenarios (pseudo-Hermiticity and $P T$-symmetric), depending on the parameter combination, the spectra of $H(γ,α,N)$ can be real even when it is non-Hermitian. Then we demonstrate, for both setups, that there is a well-defined sector of the $γα$-plane (which grows with $N$⁠) where the spectrum of $H(γ,α,N)$ is predominantly real.