Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Rank–three matroids are Rayleigh David G. Wagner. | Rank-three matroids are Rayleigh David G. Wagner Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada dgwagner@ .ca Submitted Apr 7 2004 Accepted May 11 2005 Published May 16 2005 Mathematics Subject Classifications 05B35 60C05 Abstract A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities. 1 Introduction. For the basic concepts of matroid theory we refer the reader to Oxley s book 5 . A linear resistive electrical network can be represented as a graph G V E together with a set of positive real numbers y ye e 2 E that specify the conductances of the corresponding elements. In 1847 Kirchhoff 3 determined the effective conductance of the network measured between vertices a b 2 V as a rational function Yab G y of the conductances y. This formula can be generalized directly to any matroid. For electrical networks the following property is physically intuitive if yc 0 for all c 2 E then for any e 2 E @ -@Yab G y 0. @ye That is by increasing the conductance of the element e we cannot decrease the effective conductance of the network as a whole. This is known as the Rayleigh monotonicity property. Informally a matroid has the Rayleigh property if it satisfies inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. While there are non-Rayleigh matroids of rank four or more we show here that every matroid of rank at most three is Rayleigh answering a question left open by Choe and Wagner 1 . Research supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N8 1 Let M be a matroid with ground-set E and fix indeterminates y ye e 2 E indexed by E. For a basis B of M let yB ỊỊ eeB ye and let M y BeM yB with the sum
