On Join Graph Of Zero-Divisor Graphs Of Direct Product Of Finite Fields

I. Beck introduced the  concept  of Zero-divisor  graph  of a commutative ring  R  with  all the  elements  of ring  R  as vertices  and  two  distinct  vertices x, y are adjacent if and  only if x · y = 0.  Anderson  and  Livingston  modified the  definition  of Zero-divisor  graph  given by Beck,  by considering  only the non-zero  zero-divisors  as the  vertices  of the  Zero-divisor  graph  denoted  by Γ(R)  and  two  distinct   vertices  x, y  are  adjacent  if and  only  if x · y  =  0. The  Join  graph  G + H  of two graphs  G and  H  is the  graph  with  vertex  set V (G + H) = V (G) ∪ V (H)  and  edge set E(G + H) = E(G)  ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}.  In this paper  we determine  the graph  properties  such as diameter,   girth,  clique number,  vertex  chromatic  number,  independence number  of Join graph  of Zero-divisor graphs  of direct  product  of finite fields.