Sub-Graphs of Complete Graph.

Abstract: In the complete graph K2m+3 for m ≥ 2, we study some structures of simple non-isomorphic Hamiltonian sub-graphs of the form H ( 2m+3, 6m+3) for m ≥ 2. The various structures of the form H(2m+3, 6m+3) for m ≥ 2 are found and some of them are observed to give some forms of metal atom cluster compound of chemistry. Finally, we establish an algorithm to solve the traveling salesman problem when the weights of edges of the complete graph K 2m+3 for m ≥ 2 are non-repeated in the case of symmetrical problems. The applications of the sub-graphs H (2m+3,6m+3) for m ≥ 2 are shown in the solution of traveling salesman problem. Keywords- Complete graph, Hamiltonian simple sub-graph, Non-isomorphic graph, cluster compounds, Tsp.