Topological indices of join graph of zero divisor graphs of direct product of three finite fields

Abstract

In this paper, we calculate the Weiner index, some degree-based topological indices, and some eccentricity-based topological indices of Join graph of Γ(F₁ ×F₂ ×F₃) and Γ(J₁ ×J₂ ×J₃), where F₁, F₂, F₃, J₁, J₂, J₃ are finite fields of order at least two.  The vertex set of Γ(F₁×F₂×F₃) (Γ(J₁×J₂ ×J₃)) is Z^∗(F₁ ×F₂ ×F₃) (Z^∗(J₁ ×J₂ ×J₃)), the set of non-zero zero divisors, and two distinct vertices (x₁, x₂, x₃), (y₁, y₂, y₃) are adjacent if and  only  if  (x₁, x₂, x₃) · (y₁, y₂, y₃)  = (0, 0, 0), the additive identity of the ring F₁ × F₂ × F₃ (J₁ × J₂ × J₃). The vertex set of Join graph Γ(F₁ × F₂ × F₃) + Γ(J₁ × J₂ × J₃) is V (Γ(F₁ × F₂ × F₃)) ∪ V (Γ(J₁ × J₂ × J₃)) and edge set is E(Γ(F₁ × F₂ × F₃))   ∪  E(Γ(J₁ × J₂ × J₃))  ∪  {(x₁, x₂, x₃) · (y₁, y₂, y₃) : (x₁, x₂, x₃) ∈ V (Γ(F₁ × F₂ × F₃)), (y₁, y₂, y₃) ∈ V (Γ(J₁ × J₂ × J₃))}.