Spanning Simplicial Complexes of Uni-Cyclic Graphs

Abstract

In this paper, we introduce the concept of spanning simplicial complexes ∆ s (G) associated to a simple finite connected graph G. We give the characterization of all spanning trees of the uni-cyclic graph U n,m . In particular, we give the formula for computing the Hilbert series and h-vector of the Stanley Riesner ring k ∆ s (U n,m ) . Finally, we prove that the spanning simplicial complex ∆ s (U n,m ) is shifted hence ∆ s (U n,m ) is shellable.

SPANNING SIMPLICIAL COMPLEXES OF n-CYCLIC GRAPHS WITH A COMMON VERTEX

“…In this section, we will give a formula for f -vector of ∆ s (G l1, l2, ··· , ln ) and consequently a formula for Hilbert series of the Stanley-Reisner ring k[∆ s (G l1, l2, ··· , ln )] under the assumption that the length of every cycle G li is t for 1 ≤ i ≤ n. But before this we need the following proposition, its proof can be seen in Proposition 2.2 of [1].…”

“…The note of spanning simplicial complex ∆ s (G) on edge set E of a graph G = G(V, E) was introduced in [1], the set of its facets is exactly the edge set s(G) of all possible spanning trees of G, i.e. ∆ s (G) = F i | F i ∈ s(G) .…”

“…Note that for a graph G, the problem of finding s(G) is not always easy to handle. Anwar, Raza and Kashif [1] proved some algebraic and combinatorial properties of spanning simplicial complex of the uni-cyclic graph U n (i.e., if the vertex set of U n is V = {x 1 , . .…”

SPANNING SIMPLICIAL COMPLEXES OF n-CYCLIC GRAPHS WITH A COMMON EDGE

“…In this section, we will give a formula for f -vector of ∆ s (G t1, t2, ··· , tn ) and consequently a formula for Hilbert series of the Stanley-Reisner ring k[∆ s (G t1, t2, ··· , tn )] under the assumption that the length of every cyclic graph G ti is t for 1 ≤ i ≤ n. But before this we need the following proposition, its proof can be seen in Proposition 2.2 of [1].…”

“…The note of spanning simplicial complex ∆ s (G) on edge set E of a graph G = G(V, E) was introduced in [1], the set of its facets is exactly edge set s(G) of all possible spanning trees of G, i.e. ∆ s (G) = F i | F i ∈ s(G) .…”