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We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.
For a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u eq w$ and $u,w$ are not adjacent. Then $Gamma$ is said to be $2$-geodesic transitive if its automorphism group is transit
A graph is said to be {em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive n
Let $Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $Gamma$. Fix a base vertex $xin X$ and for $0 leq i leq D$ let $E^*_i=E^*_i(x)$ denote the projection matrix to th
A graph $Gamma$ is said to be symmetric if its automorphism group $rm Aut(Gamma)$ acts transitively on the arc set of $Gamma$. In this paper, we show that if $Gamma$ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting
A graph $G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {em bi-Cayley graph/} over $H$. Such a graph $G$ is called {em normal/} if $H$ is normal in the full automorphism group of $G