Can I save the relationship map as a picture?

Yes, from version 1.9.22. Open the Codex from the book's page and choose Relations. Under the map, press "Download PNG". BookyAI saves the whole map as a high-resolution picture with the book's title on top, even the parts you have zoomed away from. The colors show the kind of each tie (the legend under the map explains them), a thicker line means a stronger tie and a dashed line a secret relationship. Drag the map to move around, and use the plus, minus and reset buttons to zoom.