The Embedding of Line Graphs Associated to the Annihilator Graphs of Commutative Rings

The annihilator graph Formula: see text of the commutative ring Formula: see text is an undirected simple graph whose vertex set is the set of non-zero zero-divisors of Formula: see text, and two distinct vertices Formula: see text and Formula: see text are adjacent if and only if Formula: see text. In this paper, we study the embedding of the line graph of Formula: see text into orientable or non-orientable surfaces. We completely characterize the finite commutative rings such that the line graph of Formula: see text is of genus or crosscap at most two. We also obtain the inner vertex number of the line graph of Formula: see text. Finally, we classify all the finite commutative rings such that the book thickness of certain line graphs of Formula: see text is at most four.