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Foster's Theorems

Let (Omega)_(ij) be the resistance distance matrix of a connected graph G on n nodes. Then Foster's theorems state that

sum_((i,j) in E(G)))Omega_(ij)=n-1,

where E(g) is the edge set of G, and

sum(Omega_(ij))/(delta_(ij))=n-2,

where the latter sum runs over all pairs of adjacent edges (i,k),(j,k) and delta_(ij) is the vertex degree of the vertex k common to those edges (Palacios 2001).

See also

Graph Distance Matrix, Resistance Distance

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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.

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Foster's Theorems

Cite this as:

Weisstein, Eric W. "Foster's Theorems." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FostersTheorems.html

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