Local Properties of the Total Graph T(Γ(Zn))

Abstract

Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with vertex set R and two distinct vertices x and y are adjacent if x+y ∈ Z(R), where Z(R) is the set of all zero divisors of R. This paper presents a study of some local properties of the graph T (Γ(Z_n)). We answer the question “ when is T (Γ(Z_n)) locally connected?”. We also prove that the neighborhoods of any two distinct vertices in T (Γ(Z_n)) induce isomorphic graphs if and only if n is even.

Key words and phrases. Total graph of a commutative ring, the ring of integers modulo n, locally onnected graph, locally homogeneous graph, local property of a graph, neighborhood of a vertex.

2010 Mathematics Subject Classification. 13A15, 05C99