Automorphisms Of Some Graphs Over Some Rings

In this dissertation,we investigate the automorphisms of some graphs over some rings.For a graph G,let σ be a bijection on the vertex set(denoted by V(G)),if σ preserves adjacency,then it is an automorphism of the graph G.Under the usual composition of functions,all auto-morphisms of graph G forms a group which is the automorphism group of the graph G,written as Aut(G).For a ring S,the zero-divisor graph of S,written as ΓZ(S),is a directed graph whose vertices are nonzero zero-divisors of S,and there is a directed edge from a vertex A to a vertex B if and only if AB=0.For a ring S,the commuting graph ΓC(S)of S is the graph associated to S whose vertices are non-central elements(S\C(S))in S,and distinct vertices A and B are adjacent if and only if AB=BA.We first completely determine the automorphisms of the zero-divisor graph of M2×2(Zps),where M2×2(2ps)is the 2 × 2 matrix ring over Zps,Zps is the ring of integers modulo ps,p is a prime and s(≥1)is a positive integer.Then we study the automorphism group of the commuting graph of M2,2(ZPs).Let Zps[i]be the ring of Gaussian integers modulo ps,where p is prime and s is a positive integer.In the last chapter,we show the automorphism groups of the unitary Cayley graph Gzps[i],the unit graph G(Zps[i])and the total graph T(Γ(Zps[i])).