We investigate when different graphs associated to commutative rings are chordal. In particular, we characterize commutative rings R with each of the following conditions: The total graph of R is chordal, the total dot product or the zero-divisor dot product graph of R is chordal, the comaximal graph of R is chordal, R is semilocal and the unit graph or the Jacobson graph of R is chordal. | Turk J Math (2018) 42: 2202 – 2213 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Chordality of graphs associated to commutative rings Ashkan NIKSERESHT∗, Department of Mathematics, Shiraz University, Shiraz, Iran Received: • Accepted/Published Online: • Final Version: Abstract: We investigate when different graphs associated to commutative rings are chordal. In particular, we characterize commutative rings R with each of the following conditions: the total graph of R is chordal; the total dot product or the zero-divisor dot product graph of R is chordal; the comaximal graph of R is chordal; R is semilocal; and the unit graph or the Jacobson graph of R is chordal. Moreover, we state an equivalent condition for the chordality of the zero-divisor graph of an indecomposable ring and classify decomposable rings that have a chordal zero-divisor graph. Key words: Chordal graph, zero-divisor graph, total graph, Jacobson graph, unit graph, comaximal graph, dot product graphs 1. Introduction In this paper all rings are commutative with identity and R denotes a ring. Recently many researchers have tried to study the algebraic structure of R by associating some graphs to R , such as zero-divisor graphs, total graphs, or unit graphs; see [1–4, 6, 9–11, 16–19]. The interrelation of graph theoretic properties of these graphs and the algebraic structure of R has been the focus of research on this topic. In particular, many have tried to find graph theoretic invariants of these graphs, such as diameter, girth, and chromatic number, from the algebraic structure of R . Some have also investigated when these graphs have some specific graph theoretic properties, such as being connected, bipartite, or Eulerian. On the other hand, some algebraic properties and invariants of R can be found from these graphs. For example, it is proved that if R and S are two finite reduced .
