The chemical potential for a novel intrinsic graph metric, the resistance distance, is briefly recalled, and then a number of »sum rules« for this metric are established. »Global« and »local« types of sum rules are identified. The sums in the »global« sum rules are graph invariants, and the sum rules provide inter-relations amongst different invariants, some involving the resistance distance while others do not. Illustrative applications to more »regular« graphs are made.
