Characterisations and Structural Properties of Pure n-Simplicial Trees

This paper extends the concept of trees in graphs to the context of pure n-simplicial complexes by generalising the notion of paths and cycles (in graphs). We introduce and study these higher-dimensional analogues of trees known as pure n-simplicial trees. Our main result is to establish the equivalence between pure n-simplicial trees and the (m, n)-trees introduced by Dewdney in 1974 when m = n - 1, thereby extending and improving his original characterisation. We also disprove two conjectures proposed by Dewdney in 1974 by providing counterexamples, and we present a refined version of the conjecture with an added condition, along with a rigorous proof. Our results contribute to the theoretical knowledge of higher-dimensional trees and their structural properties.