A New Proof of Foster's First Theorem. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Zhang, Jingyuan; Yan, Weigen
• Is Part Of: American Mathematical Monthly. Volume 127:Issue 1(2020)
• Page Start: 72
• Page End: 74
• Publication Date: 2020-01-02
• Subjects: MSC: Primary 05C30 -- Secondary 05C81; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2020.1668709
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: Let G be an electrical network graph with vertex set V and edge set E, and let Ω ( i, j ) be the effective resistance between vertices i and j of G . Foster's first theorem states that ∑ ( i, j ) ∈ E c i j Ω ( i, j ) = | V | − 1, where cij is the conductance of the edge ( i, j ). In this note, we give a combinatorial proof of two equivalent formulae on the weighted enumeration of spanning trees of an edge-weighted graph, one of which results in a new proof of Foster's first theorem.