Square element graph of square-subtract rings

Abstract

If [Formula: see text] is a ring then the square element graph [Formula: see text] is the simple undirected graph whose vertex set consists of all non-zero elements of [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text] for some [Formula: see text]. In this paper, we provide some necessary and sufficient conditions for the connectedness of [Formula: see text], where [Formula: see text] is a ring with identity. We mainly characterize some special class of ring [Formula: see text] which we call square-subtract ring for which the graph [Formula: see text] is connected.