# Graphs that are vertex transitive but not edge transitive

A graph $G$ is vertex transitive if for any two vertices $x$ and $y$ of $G$ there is an automorphism of $G$ that sends $x$ to $y$. Similarly, $G$ is edge transitive if for any two edges $e$ and $f$ of $G$ there is an automorphism of $G$ that sends $e$ to $f$.

A necessary condition for vertex transitivity is that $G$ be regular. If $G$ is regular and edge transitive but not vertex transitive, it is called semi-symmetric.

1. Is there a name for graphs that are vertex transitive but not edge transitive? Is there a characterization for these types of graphs?

2. Sometimes a graph can be as close as possible to being edge transitive without actually having that quality, ie. the edge automorphism group of $G$ (the automorphism group of the line graph of $G$) has exactly two orbits. If this is the case does it imply anything interesting about $G$ (besides that there are essentially two "types" of edges in $G$)? I'm particularly interested in the case where $G$ is vertex transitive.

Thank you.

The graph corresponding to a triangular prism is vertex-transitive but not edge-transitive. More generally, two copies $G,G'$ of the complete graph $K_n$ for $n\geq 3$ that are linked by $n$ edges (one edge between $v_i$ and $v_i'$ for each $i$) is vertex-transitive but not edge-transitive.