The combinatorics of algebraic graph theory in theoretical physics

Abstract

For many years physicists have been using graphs to represent the combinatorial properties of complicated algebraic expressions by defining sets of graphs whose generating functions are the physical quantities under investigation. In statistical mechanics there has recently been a trend towards the use of transformations which reduce the complexity of the graphical enumeration at the expense of an increase in algebraic complexity. A combinatorial analysis of algorithms used in actual computations shows why such a ‘trade-off’ is desirable. Series expansions for the limit of chromatic polynomials and for lattice models in statistical mechanics are considered as examples.