The NF-Number of Two Complete Graphs Joined by a Common Vertex

Abstract

Let $$\Delta$$ be a simplicial complex on the vertex set $$V$$. For $$m=1,2,3,\dots$$, the notion of $$m$$-th $$\mathcal{NF}$$-complex of $$\Delta$$, $$\delta^{(m)}_{\mathcal{NF}}(\Delta)$$, was introduced by Hibi and Mahmood in [5], where $$\delta^{(m)}_{\mathcal{NF}}(\Delta)=\delta_{\mathcal{NF}}(\delta^{(m-1)}_{\mathcal{NF}}(\Delta))$$ with setting $$\delta^{(1)}_{\mathcal{NF}}(\Delta)=\delta_{\mathcal{NF}}(\Delta)$$ such that $$\delta_{\mathcal{NF}}(\Delta)$$ is the Stanley–Reisner complex of the facet ideal of $$\Delta$$. The $$\mathcal{NF}$$-number of $$\Delta$$ is the least positive integer $$q$$ for which $$\delta^{(q)}_{\mathcal{NF}}(\Delta)\simeq\Delta$$. In this paper, we investigated the $$\mathcal{NF}$$-number of two copies of complete graphs $$K_n$$ joined by one common vertex $$\{u\}$$. At the end, we also provided an explicit example for the case of two copies of $$K_5$$ joined by common vertex $$\{u\}$$.