T. L. Hill's Graphical Method for Solving Linear Equations

That tree enumeration can be used for solving systems of linear equations was independently realized by physicists [9], electrical engineers [11], chemists [7] and most recently by biophysicists [4, 5]. Going in the opposite direction, mathematicians realized that one can use linear algebra to count trees ([1, p. 378; 8, p. 578; 10, pp. 39-51]). However, both mathematicians and scientists had to resort to 'fancy' mathematics in order to establish this connection. It is therefore remarkable that the biophysicist T. L. Hill came up with a very elegant and short combinatorial proof [4, 5]. We are going to present Hill's proof in standard mathematical language and show how it implies the matrix-tree theorem. We also extend Hill's method to inhomogeneous equations and show how it justifies the Wang algebra of networks, which was extensively studied by Duffin [3] and Bott and Duffin [2].