The Algebraic Structures Associated to Spanning Trees

Abstract

In this thesis, we introduce the concept of spanning simplicial complex s(G) corresponding to finite simple graph G(V, E). These simplicial complexes are of worth-importance as they build a nice connection between algebra and combinatorics. We explore some algebraic invariants in between finite simple graphs and simplicial complexes arising from the spanning trees of these graphs. In particular, we characterize stanely-reisner ideals IN ( s(Cn )) of spanning simplicial complexes of cyclic graphs. For s(Cn ), we give the formulation of f-vector and h-vector. Moreover, we compute the formula to calculate the Hilbert series of k[ s(Cn)] and most importantly, we show that simplicial complexes arising from the spanning trees of cyclic graphs are shellable. We also give the formula for f-vector of spanning simplicial complex corresponding to friendship graph and computed the Hilbert series of stanely-reisner ring k[ s(Fn )] of this complex. We discuss the primary decomposition of the facet ideal of s(Fn) and give all its the associated primes. At the end, we show that the facet ideal of spanning simplicial complex of friendship graph is unmixed of height 2.