A study of unit graphs and unitary cayley graphs associated with rings

In this thesis, we study the unit graph G(R) and the unitary Cayley graph Γ(R) of a ring R, and relate them to the structure of the ring R. Chapter 1 gives a brief history and background of the study of the unit graphs and unitary Cayley graphs of rings. Moreover, some basic concepts, which are needed in this thesis, in ring theory and graph theory are introduced. Chapter 2 concerns the unit graph G(R) of a ring R. In Section 2.2, we first prove that the girth gr(G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. Then we determine the rings R with R/J(R) semipotent and with gr(G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr(G(R)) = 3 or 4. The girth of the unit graphs of some ring extensions are also investigated. The focus of Section 2.3 is on the diameter of unit graphs of rings. We prove that diam(G(R)) ∈ {1, 2, 3,∞} for a ring R with R/J(R) self-injective and determine those rings R with diam(G(R)) = 1, 2, 3 or ∞, respectively. It is shown that, for each n ≥ 1, there exists a ring R such that n ≤ diam(G(R)) ≤ 2n. The planarity of unit graphs of rings is discussed in Section 2.4. We completely determine the rings whose unit graphs are planar. In the last section of this chapter, we classify all finite commutative rings whose unit graphs have genus 1, 2 and 3, respectively. Chapter 3 is about the unitary Cayley graph Γ(R) of a ring R. In Section 3.2, it is proved that gr(Γ(R)) ∈ {3, 4, 6,∞} for an arbitrary ring R, and that, for each n ≥ 1, there exists a ring R with diam(Γ(R)) = n. Rings R with R/J(R) self-injective are classified according to diameters of their unitary Cayley graphs. In Section 3.3, we completely characterize the rings whose unitary Cayley graphs are planar. In Section 3.4, we prove that, for each g ≥ 1, there are at most finitely many finite commutative rings R with genus γ(Γ(R)) = g. We also determine all finite commutative rings R with γ(Γ(R)) = 1, 2, 3, respectively. Chapter 4 is about the isomorphism problem between G(R) and Γ(R). We prove that for a finite ring R, G(R) ∼= Γ(R) if and only if either char(R/J(R)) = 2 or R/J(R) = Z2 × S for some ring S.