# When the lexicographic product of two graphs is edge transitive?

A graph $G$ is said to be edge transitive provided that, for any two edges $f$ and $g$ in $G$ , there is an automorphism of $G$ sending $f$ to $g$.

Let $G$ and $H$ be two graphs on vertex sets $V(G)$ and $V(H)$, respectively. Then their lexicographic product $G\circ H$ is a graph denoted by $V(G\circ H)=V(G) \times V(H)$, and there is an edge from $(u,x)$ to $(v,y)$ if either $u=v$ and $x$ be adjacent to $y$ in $H$, or $u$ be adjacent to $v$ in $G$.

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