The Zariski topology graph on scheme

Abstract

Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.