On Realizability of a Set of Trees

A matrix with entries O's and I's whose rows correspond to the trees of a graph G and whose columns correspond to the elements of G is introduced. Many interesting properties of such a matrix are derived. The concept of the rank (modulus 2) of the tree matrix is found to be very useful in determining the number of separable parts of the corresponding graph. A simple algebraic way is presented by which one can find, from a given set of trees (or tree matrix), the fundamental circuit (or cut-set) matrix with respect to a prespecified tree. It is also shown that one can easily find, from the tree matrix, the set of all paths between the vertices of an element of a graph. Some interesting conjectures are stated concerning graphs with a given rank and nullity which have a minimum number of trees.