# Automorphism groups of bipartite direct products

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Richard Hammack

Virginia Commonwealth University

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**Minisymposium:**
GRAPH PRODUCTS

**Content:**
We are concerned with automorphism groups of thin direct products. (A graph is {\it thin}
if no two vertices have the same neighborhood.)
It is known that if $\varphi$ is an automorphism of a connected non-bipartite thin graph $G$
that factors (uniquely) into primes as $G=G_1\times G_2\times\cdots\times G_k$,
then there is a permutation $\pi$ of $\{1,2,\ldots,k\}$, together with
isomorphisms $\varphi_i:G_{\pi(i)}\to G_i$, such that
$$\;\varphi(x_1,x_2,\ldots,x_k)=\left(\,\varphi_1(x_{\pi(1)}),\,\varphi_2(x_{\pi(2)}),\,\ldots,\,\varphi_k(x_{\pi(k)})\,\right).$$
This talk explores an analogous result for {\it bipartite} graphs $G$. Unlike the non-bipartite
case, such graphs do not factor uniquely into primes. But if
$G=A\times B$, where $B$ is prime and bipartite, then $B$ is unique up to isomorphism,
though
the non-bipartite factor $A$ is not uniquely determined.
We codify the structure of automorphisms of thin connected bipartite products $A\times B$,
where $B$ is prime and bipartite.