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Given two (simple, undirected, finite) graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$, let their automorphism groups be $Aut(G_1)$ and $Aut(G_2)$.

I'll recall that the cartesian product $G_1 \times G_2$ has vertex set $V_1 \times V_2$ , and two vertices $(a,b) , (x,y) \in V_1 \times V_2$ are adjacent iff $(a,x) \in E_1$ and b=y, or $(b,y) \in E_2$ and a=x.

My question is: does the problem of determining $Aut(G_1 \times G_2)$ in terms of $Aut(G_1)$ and $Aut(G_2)$ has a simple answer? May you suggest some bibliographic reference about this and related problems? Basic texts about graph theory usually barely define the automorphism group, and more algebraically-oriented texts i found did not wuite answered the question.

I tried to find a way to answer the question by myself, but i did not succeed. I'd like to know if the problem is really not-so-trivial, or if i'm simply not smart enough :) Thanks in advance for any comment.

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# Automorphism group of the cartesian product of two graphs.

Given two (simple, undirected, finite) graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$, let their automorphism groups be $Aut(G_1)$ and $Aut(G_2)$.

I'll recall that the cartesian product $G_1 \times G_2$ has vertex set $V_1 \times V_2$ , and two vertices $(a,b) , (x,y) \in V_1 \times V_2$ are adjacent iff $(a,x) \in E_1$ or $(b,y) \in E_2$.

My question is: does the problem of determining $Aut(G_1 \times G_2)$ in terms of $Aut(G_1)$ and $Aut(G_2)$ has a simple answer? May you suggest some bibliographic reference about this and related problems? Basic texts about graph theory usually barely define the automorphism group, and more algebraically-oriented texts i found did not wuite answered the question.

I tried to find a way to answer the question by myself, but i did not succeed. I'd like to know if the problem is really not-so-trivial, or if i'm simply not smart enough :) Thanks in advance for any comment.