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

Patil, Kailash M.

doi:10.47974/JDMSC-1909

Open Access Research Article

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

^*Kailash M. PatilCorresponding authorkailashpatil2111.maths@ddu.ac.in kailashpatil2111@gmail.comDepartment of MathematicsFaculty of TechnologyDharmsinh Desai UniversityNadiad, Gujarat, 387001, IndiaView full profile →

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pp. 639–650Vol. 27Issue 2-BMarch 2024DOI: 10.47974/JDMSC-1909 Crossmark XML

Published Online:: 11 Apr 2024
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-1909
Pages:: 639–650

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.

Keywords

Cayley Hamilton theorem Trace of a matrix Powers of a matrix Directed graph Adjacency matrix

Subject Classifications

Primary 15A24Secondary 15A45

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