Some applications of trace of powers of matrices to graph theory, algebra and combinatorics

Abstract

In this paper, using Cayley Hamilton Theorem, we exhibit a combinatorial formula for Tr(An), (n ∈ ℤ, A ∈M3 (K)), over the field K(= ℝorℂ)using Tr (A), det(A) and sum of principal minors of A. Further, a formula for ∑nk=0 Tr(Ak) is derived. For specific values of Tr(A) and det(A), the formula of Tr(An) is highly simplified. We exhibit some applications of our results in Graph Theory, Algebra and Combinatorics. We have verified our results on various matrices using Scilab and found that Scilabfails tremendously because of the truncation. But our formulae give exact result.