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Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prismis the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and itscomplement $\bar{\Gamma}$ by adding a perfect matching, where each its edgeconnects two copies of the same vertex in $\Gamma$ and $\bar{\Gamma}$. Itgeneralizes the Petersen graph, which is obtained if $\Gamma$ is the pentagon.The automorphism group of $\Gamma\bar{\Gamma}$ is described for arbitrary graph$\Gamma$. In particular, it is shown that the ratio between the cardinalitiesof the automorphism groups of $\Gamma\bar{\Gamma}$ and $\Gamma$ can attain onlyvalues $1$, $2$, $4$, and $12$. It is shown that the Cheeger number of$\Gamma\bar{\Gamma}$ equals either 1 or $1-\frac{1}{n}$, and the twocorresponding classes of graphs are fully determined. It is proved that$\Gamma\bar{\Gamma}$ is vertex-transitive if and only if $\Gamma$ isvertex-transitive and self-complementary. In this case the complementary prismis Hamiltonian-connected whenever $n>5$, and is not a Cayley graph whenever$n>1$. The main results involve endomorphisms of graph $\Gamma\bar{\Gamma}$. Itis shown that $\Gamma\bar{\Gamma}$ is a core, i.e. all its endomorphisms areautomorphisms, whenever $\Gamma$ is strongly regular and self-complementary.The same conclusion is obtained for many vertex-transitive self-complementarygraphs. In particular, it is shown that if there exists a vertex-transitiveself-complementary graph $\Gamma$ such that $\Gamma\bar{\Gamma}$ is not a core,then $\Gamma$ is neither a core nor its core is a complete graph.

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs