Combinatorics and Graph Theory


The Complete Graph: Eigenvalues, Trigonometrical Unit-Equations with Associated T-Complete-Eigen Sequences, Ratios, Sums and Diagrams

Authors: Paul August Winter, Carol Lynne Jessop, Fadekemi Janet Adewusi

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 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 general determinant equation involving the minor of the matrix associated with this characteristic polynomial.

Comments: 20 Pages.

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[v1] 2014-11-19 05:12:27

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