# On Algebraic Characterization of SSC of the Jahangir's Graph $\mathcal{J}_{n,m}$

## TL;DR

This paper provides algebraic and combinatorial characterizations of the spanning simplicial complex of Jahangir's graph, including formulas for its f-vector, Hilbert series, and Cohen-Macaulay property.

## Contribution

It introduces new algebraic and combinatorial descriptions of the simplicial complex associated with Jahangir's graph, including formulas and properties not previously established.

## Key findings

- The simplicial complex is pure.
- Derived the formula for the f-vector.
- Proved the face ring is Cohen-Macaulay.

## Abstract

In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex $\Delta_s(\mathcal{J}_{n,m})$ of the Jahangir's graph $\mathcal{J}_{n,m}$ are explored. We show that $\Delta_s(\mathcal{J}_{n,m})$ is pure, present the formula for $f$-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring $k[\Delta_s(\mathcal{J}_{n,m})]$. Finaly, we show that the face ring of $\Delta_s(\mathcal{J}_{n,m})$ is Cohen-Macaulay and give some open scopes of the current work.