Graphical properties of the bipartite graph of $\operatorname{Spec}(\mathbb{Z}[x]) \setminus \{0\}$

Abstract

Consider $\operatorname{Spec}(\mathbb{Z}[x])$, the set of prime ideals of $\mathbb{Z}[x]$ as a partially ordered set under inclusion. By removing the zero ideal, we denote $G_{\mathbb{Z}}=\operatorname{Spec}(\mathbb{Z}[x]) \setminus \{0\}$ and view it as an infinite bipartite graph with the prime ideals as the vertices and the inclusion relations as the edges. In this paper, we investigate fundamental graph theoretic properties of $G_{\mathbb{Z}}$. In particular, we describe the diameter, circumference, girth, radius, eccentricity, vertex and edge connectivity, and cliques of $G_{\mathbb{Z}}$. The complement of $G_{\mathbb{Z}}$ is investigated as well.

Received: 3 August 2014 | Accepted: 30 December 2014