Automorphism groups of bipartite direct products

Richard Hammack
Virginia Commonwealth University


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.

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