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.