Let $Gamma$ be a finite simple graph with automorphism group $Aut(Gamma).$ An automorphism $sigma$ of $Gamma$ is said to be an adjacency automorphism, if for every vertex $xin V(Gamma)$, either $sigma x=x$ or $sigma x$ is adjacent to $x$ in $Gamma$. A shift is an adjacency automorphism fixing no vertices. The graph $Gamma$ is (shift) adjacency-transitive if for every pair of vertices $x, x'in V(Gamma)$, there exists a sequence of (shift) adjacency automorphisms $sigma_{1},sigma_{2},...sigma_{k}inAut(Gamma)$ such that $sigma_{1}sigma_{2}...sigma_{k}x=x'$. If, in addition, for every pair of adjacent vertices $x, x'in V(Gamma)$ there exists an (shift) adjacency automorphism say $sigmain Aut(Gamma)$ sending $x$ to $x'$, then $Gamma$ is strongly (shift) adjacency-transitive. If for every pair of adjacent vertices $x, x'in V(Gamma)$ there exists exactly one shift $sigmainAut(Gamma)$ sending $x$ to $x'$, then $Gamma$ is uniquely shift-transitive. In this paper, we investigate these concepts in some standard graph products.
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