THE COMPLETE PRODUCT OF TWO FUZZY GRAPHS AND ITS RELATIONSHIP WITH FUZZY GRAPH ISOMORPHISM

Fuzzy graph was introduced by Kaufmann [7] in 1973. In this paper, we introduced the concept of the complete product of two fuzzy graphs with an Illustrative example. We proved the result that If $G:(\sigma,\mu)=(U,E_U)$, $H:(\tau,\vartheta)=(V,E_V)$, $G':(\sigma',\mu')=(U',E_{U'})$ and $H':(\tau',\vartheta')=(V',E_{V'})$ are any four fuzzy graphs such that $G:(\sigma,\mu)\cong G':(\sigma',\mu')$ and $H:(\tau,\vartheta)\cong H':(\tau',\vartheta')$ under the fuzzy graph isomorphisms $f$ and $h$ respectively, then $G\times_PH\cong G'\times_P H'$. As the proof is too long, we have demonstrated the result in two by parting into two hypotheses.