The Line Graphs Arising From Graphs Associated to Rings: A Survey with New Results

The algebraic graph theory is concerned with the interplay between algebra and graph theory. Many interesting results can be derived by associating a graph to an algebraic structure. Given a graph [Formula: see text] with at least one edge, the line graph [Formula: see text] is that graph whose vertices are the edges of [Formula: see text], with two of these vertices being adjacent if the corresponding edges are adjacent in [Formula: see text]. In this paper, we highlight the rich interplay between the two topics viz line graphs and graphs from rings. In the last decade, extensive contribution has been made towards the investigation on these topics. Our main motive is to provide a complete survey on the line graphs arising from the unit graphs of rings, the total graphs of rings, the Von Neumann regular graphs of rings since from their development as well as to investigate some properties of the unit line graphs associated with rings and the line graphs associated with [Formula: see text]-graphs. Further, we gave the complete survey of line signed graph arising from signed unit graph and signed total graph of commutative rings.