Simultaneous localization and mapping (SLAM) in robotics, and a number of related problems that arise in sensor networks are instances of estimation problems over weighted graphs. This paper studies the relation between the graphical representation of such problems and estimationtheoretic concepts such as the Cramér-Rao lower bound (CRLB) and D-optimality. We prove that the weighted number of spanning trees, as a graph connectivity metric, is closely related to the determinant of CRLB. This metric can be efficiently computed for large graphs by exploiting the sparse structure of underlying estimation problems. Our analysis is validated using experiments with publicly available pose-graph SLAM datasets.