Foster's Theorems

Let  be the resistance distance matrix of a connected graph  on  nodes. Then Foster's theorems state that

where  is the edge set of , and

where the latter sum runs over all pairs of adjacent edges  and  is the vertex degree of the vertex  common to those edges (Palacios 2001).

REFERENCES:

Foster, R. M. "The Average Impedance of an Electrical Network." In Contributions to Applied Mechanics (Reissner Anniversary Volume). Ann Arbor, MI: Edwards Brothers, pp. 333-340, 1949.

Foster, R. M. "An Extension of a Network Theorem Contributions to Applied Mechanics." IRE Trans. Cir. Th. 8, 75-76, 1961.

Klein, D. J. and Randić, M. "Resistance Distance." J. Math. Chem 12, 81-95, 1993.

Palacios, J. L. "Closed-Form Formulas for Kirchhoff Index." Int. J. Quant. Chem. 81, 135-140, 2001.

Tetali, P. "Random Walks and the Effective Resistance of Networks." J. Theor. Prob. 4, 101-109, 1991.

Tetali, P. "An Extension of Foster's Network Theorem." Combin. Prob. Comp. 3, 421-427, 1994.

Weinberg, L. "Kirchhoff's 'Third and Fourth Laws." IRE Trans. Cir. Th. 5, 8-30, 1958.