Let [K.sub.n] be the

complete graph of order n; then [zeta]([K.sub.n]) = n, [[xi].sup.c]([K.sub.n]) = n(n - 1), [M.sup.*.sub.1] ([K.sub.n]) = n, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the total number of edges of the elements of is Hence, Now the elements of form a

complete graph of type So total edges of the vertices of are and each element of is adjacent to all elements of .

A

complete graph with vertices n[epsilon]N, denoted by [K.sub.n] is a connected simple graph with every vertex is connected with every other vertex by an edge.

For any connected graph G with p vertices p [greater than or equal to] 2, [[gamma].sub.M](G) = p - [lambda](G), where [lambda](G) is a edge connectivity of G if and only if G is a

complete graph of order p.

Up to now, the spectra are known for some particular graphs: path, cycle,

complete graph, complete bipartite graph, complete tree, hypercube, k-dimensional lattice, star graph, etc.

We denote a

complete graph of n vertices by [K.sub.n] and denote a star graph with k leaves by [S.sub.k], where the leaf refers to the vertex of degree one.

A variant of the traveling salesman problem called MaxLTSP, where the goal is to compute a Hamiltonian tour of maximum number of colors in an edge-colored

complete graph, has been considered in [4, 10].

The

complete graph [K.sub.n] on n vertices is referred to as an n-clique.

Let [K.sub.n], [C.sub.n], [Q.sub.m], [[product].sub.m], [A.sub.m] denote the

complete graph with n vertices, the cycle on n vertices, m-dimensional hypercube, m-sided prism, and the m-sided antiprism, respectively.

As usual, [P.sub.n], [C.sub.n], and [K.sub.n] stand for the path, the cycle, and the

complete graph with n vertices, respectively.

By proceeding in an iterative way according to claim 3.2, it is possible to effectively find an ordering of the pairs [V(G).sup.(2)] - E([PSI](G)) such that the tree T may be realized as a decomposition tree of the

complete graph [K.sub.V(G)].

For a connected graph G, N(G) [congruent to] G if, and only if, G is either a

complete graph or an odd cycle of order [greater than or equal to] 3.