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Some properties of transposition graphs

For every finite graph G without isolated vertices, there is an associated set of transpositions ((G) which correspond in a natural way to the edges of G. L1(G)s generates a group H which is a symmetric group iff G is connected. The Cayley graph f H,11) clearly depends only on G, and is called the transposition graph of G,.r(G).The distance between any two vertices of a transposition graph r(G) is established in the cases where G is a complete graph, a complete graph with an edge deleted, a path graph, or a star. The diameter of r(G) is obtained as a corollary in these cases. General upper and lower bounds: are found for the diameter of r(G) which depend on the number of vertices and the diameter of G.~' If G has no connected components isomorphic to C4 or Kn then the automorphisms of ('(G) are completely determined by the automorphisms of G. In particular, if G is a connected graphon n~vertices with no nontrivial automorphisms, then t7(G) is a graphical regular represent ation of Sn. Every transposition graph with at least four vertices is hamiltonian. If the complement of the line graph of a graph G is hamiltonian then the genus of r(G) depends only on the number of vertices and edges of G. This result can be generalised if G has no circuits of length three. Finally, it is proved that the Complement of the line graph of a graph G is hamiltonian if every vertex of G is incident to at most half the edges of 0 and every edge of G is nonincident to at least two other edges of G,, provided G has at least thirty four edges.
