On the total graph and one of its extension

Thetotalgraphofacommutativering R,denotedby T(Γ(R)),isagraphwith all elements of R as vertices, and two distinct vertices x,y ∈ R are adjacent if and only if x+y ∈ Z(R), where Z(R) denotes the set of zero-divisors of R. One of total graph’s extension defined as follows. Let S be a multiplicatively closed subset of R. The graph ΓS(R) is a simple graph with vertex set R and two distinct vertices x,y ∈ R are adjacent if and only if x + y ∈ S. In this paper, we examine the preservation of the diameter, girth and planarity of T(Γ(R)) and ΓS(R) under extension to polynomial rings