THE COMPLETE GRAPH: EIGENVALUES, TRIGONOMETRICAL UNIT-EQUATIONS WITH ASSOCIATED t-COMPLETE-EIGEN SEQUENCES, RATIOS, SUMS AND DIAGRAMS

The complete graph is often used to verify certain graph theoretical definitions and applications. Regarding the adjacency matrix, associated with the complete graph, as a circulant matrix, we find its eigenvalues, and use this result to generate a novel, and general, trigonometrical unit-equations involving the sum of terms of the form , where a is odd. This gives rise to t-complete-eigen sequences and diagrams, similar to the famous Farey sequence and diagram. We show that the ratio, involving sum of the terms of the t-complete eigen sequence, converges to ½, and use this ratio to find the t-complete eigen area. To find the eigenvalues, associated with the characteristic polynomial of complete graph, using induction, we create a novel, and general, determinant equation involving the minor of the matrix associated with this characteristic polynomial.