Spanning Forest Complexes of Union of Multi Cyclic Graphs

Abstract

The objective of this thesis, is to investigate the algebraic and combinatorial aspects of spanning forest complex ∆s(G) associated to simple disconnected graph G. We pay our attention to the disjoint union of copies of multi-cyclic graphs and we keep the graph tCn,k under discussion where tCn,k is a disconnected graph i.e union of copies of multi-cyclic graph Cn,k having k cycles each of length n. We examine the total number of spanning forests of the graph tCn,k in two cases. Also we define the characterization of s(tCn,k), where s(tCn,k) is the cluster of edge sets of all the spanning forests of tCn,k. We expect to give the algebraic and combinatorial properties of spanning forest complex ∆s(tCn,k) associated to tCn,k. We determine the dimension of ∆s(tCn,k) associated to tCn,k. Particularly we integrate the formula for the f − vector of spanning simplicial complex of disconnected graph of union of copies of multi-cyclic graphs.