Springer Iranian Journal of Science and Technology (Sciences) 1028-6276 Articles in Press 2016 03 25 ADJACENCY AND SHIFT-TRANSITIVITY IN GRAPH PRODUCTS 3640 10.22099/ijsts.2016.3640 EN Mohamad A. Iranmanesh Yazd University Journal Article 2015 07 22 Let $Gamma$ be a finite simple graph with automorphism group $Aut(Gamma).$<br>An automorphism $sigma$ of $Gamma$ is said to be an adjacency automorphism, if for every vertex<br>$xin V(Gamma)$, either $sigma x=x$ or $sigma x$ is adjacent to<br>$x$ in $Gamma$. A shift is an adjacency automorphism fixing no vertices. The graph<br>$Gamma$ is (shift) adjacency-transitive if for every pair of vertices $x, x'in V(Gamma)$,<br>there exists a sequence of (shift) adjacency automorphisms $sigma_{1},sigma_{2},...sigma_{k}inAut(Gamma)$<br>such that $sigma_{1}sigma_{2}...sigma_{k}x=x'$. If, in addition, for every pair of<br>adjacent vertices $x, x'in V(Gamma)$ there exists an (shift) adjacency automorphism say<br>$sigmain Aut(Gamma)$ sending $x$ to $x'$, then $Gamma$ is strongly (shift) adjacency-transitive.<br>If for every pair of adjacent vertices $x, x'in V(Gamma)$ there exists exactly one shift $sigmainAut(Gamma)$<br>sending $x$ to $x'$, then $Gamma$ is uniquely shift-transitive. In this paper, we investigate these concepts in<br>some standard graph products.