  An exact-diagonalization technique on finite-size clusters is used to study
the ground state and excitation spectra of the two-dimensional effective
fermion model, a fictious model of hole quasiparticles derived from numerical
studies of the two-dimensional t-J model at low doping. We show that there is
actually a reasonable range of parameter values where the $d_{x^2-y^2}$-wave
pairing of holes occurs and the low-lying excitation can be described by the
picture of Bogoliubov quasiparticles in the BCS pairing theory. The gap
parameter of a size $\Delta_d\simeq 0.13|V|$ (where $V$ is the attractive
interaction between holes) is estimated at low doping levels. The paired state
gives way to the state of clustering of holes for some stronger attractions.
