The self inverse element graph over a ring

Abstract

In this paper, the self-inverse element graph S_i(R) of a commutative ring R with unity is introduced as a simple undirected graph whose vertex set is R, where two distinct vertices in R are adjacent if and only if their sum is a self-inverse element of R. Various graph-theoretic properties of this graph, including size, regularity, connectedness, girth, completeness, bipartiteness, and planarity, are established for arbitrary commutative rings. Conditions under which the graph is a path or a cycle are also obtained. Furthermore, depending on the characteristic, graphical properties of the self-inverse element graph of the Cartesian product of the rings Z_n (n ≥ 2) are examined, and the structure of the graph for finite fields is completely determined.