Line Graph Isomorphisms

The precursor of line graphs as an object of study was in isomorphisms between graphs and automorphisms of graphs. Consider a one-to-one function from the edges of a graph with three edges with a common vertex and a graph with three edges forming a cycle. As shown by Hassler Whitney, this mapping not only preserves adjacency of edges, but these two are the only connected non-isomorphic graphs with this property. Whitney also proved that there are just three connected graphs having automorphisms in which three edges forming triangles and Y-graphs get interchanged. Groups connected with line graphs are also explored in this chapter, and it concludes with an interesting result on graphs that are not only isomorphic to their line graph but with their vertex-sets being preserved.