On the Algebraic Relationships of Trees, Co-Trees, Circuits, and Cutsets of a Graph

Abstract

Algebraic relationships between trees, ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> trees) and circuits, co-trees (co- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> trees) and cutsets of a graph are presented. They may be considered as a generalization and an extension of Hakimi and Green's technique of enumerating trees and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> trees. Algebraic relationships for generating trees and co-trees by elementary transformations are also given.