The properties of the volume operator in loop quantum gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. We find that the occurrence of a smallest non-zero eigenvalue is dependent upon the geometry of the underlying graph and is not a property of the volume operator itself. The present analysis benefits from the general simplified formula for matrix elements of the volume operator derived in Brunnemann and Thiemann (2006 Class. Quantum Grav. 23 1289), making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4-valent vertices are included. This is a companion paper to Brunnemann and Rideout (2007 Properties of the volume operator in loop quantum gravity: I. Results Preprint 0706.0469), providing details of the analysis presented there.