# TRACE OF THE ADJACENCY MATRIX n×n OF THE CYCLE GRAPH TO THE POWER OF TWO TO FIVE

• Fitri Aryani Department of Mathematics, Faculty of Science and Technology, State Islamic University of Sultan Syarif Kasim Riau
• Dian Ayu Puspita Department of Mathematics, Faculty of Science and Technology, State Islamic University of Sultan Syarif Kasim Riau
• Corry Corazon Marzuki Department of Mathematics, Faculty of Science and Technology, State Islamic University of Sultan Syarif Kasim Riau
• Yuslenita Muda Department of Mathematics, Faculty of Science and Technology, State Islamic University of Sultan Syarif Kasim Riau
Keywords: cycle graph, adjacency matrix, direct proof, trace matrix

### Abstract

The main aim of this research is to find the formula of the trace of adjacency matrix  from a cycle graph to the power of two to five. To obtain the general form, the first step is finding the general formula of the adjacency matrix from a cycle graph to the power of two to five. Furthermore, the formula of the trace of adjacency matrix which is mentioned above obtained and proven by direct proof. We also present an implementation of the formula which is given by an example.

### References

Data, B.N, dan Datta, K. An Algorithm for Computing Power of a Hessenberg Matrix and its Applictions, Linear Algebra and its Application, 14, 273-284. 1976.

Chu, M.T, and Raleigh. Symbolic Calculation of the Trace of the Power of a Tridiagonal Matrix, Computing, 35, 257-268. 1985.

Pan, V. Estimating the Extremal Eigenvalues of a Symetric Matrix, Computers & Mathematics with Applications, 20, 17-22. 1990.

Zarelua, A.V. “On Congruences for the Trace of Power of Some Matrices”. Proceedings of the Steklov Institute of Mathematics, 263, 78-98, 2008.

Avron, H., Counting Triangles in Large Graphs Using Randomized Matrix Trace Estimation. Proceeding of Kdd-Ldmta’10, 2010.

Brezinski, C, P.Fika dan M.Mitrouli, Estimations of the Trace of Power of Positive by Extrapolation of the Moment, Electronic Transactions on Numerical Analysis, 39, 144-155, 2012.

Pahade, J., and M. Jha, Trace of Positive Integer Power of Real 2×2 matrices, Advancesin Linear Algebra & Matrix Theory, 5, 150-155, 2015.

Pahade, J., and M. Jha, Trace of Positive Integer Power of Adjacency Matrix, Global Journal of Pure and Applied Mathematics, Vol 13 (6), 2017

Aryani, F, dan Solihin, M. Trace Matriks Real Berpangkat Bilangan Bula Negatif, Jurnal Sains Matematika dan Statistika, Vol.3 (2), 2017.

F. Aryani, dkk, "Trace Matriks Ketetanggan n x n Berpangkat -2, -3, -4" Proceding SNTIKI 12, hal 543-553,

Desember 2020.

K. H. Rosen, Discrete Mathematics and Its Applications. New York: Mc Graw Hill, 2007

R. Munir, Matematika Disktrit Edisi Ketiga. Bandung: Informatika, 2005.

M. dan Marjono, AljabarLinear. Malang: UB Press, 2012.

H. A. dan C. Rorres, Elementary Linear Algebra. Wiley: United States of Amerika, 2013.

R. Rifa’i, Aljabar Matriks Dasar. Yogyaka: Budi Utama, 2016.

H. A. dan C. Rorres, Aljabar Linear Elementer Versi Aplikasi Edisi Kedelapan. Jakarta: Erlangga, 2004

Published
2022-06-01
How to Cite
[1]
F. Aryani, D. Puspita, C. Marzuki, and Y. Muda, “TRACE OF THE ADJACENCY MATRIX n×n OF THE CYCLE GRAPH TO THE POWER OF TWO TO FIVE”, BAREKENG: J. Math. & App., vol. 16, no. 2, pp. 393-408, Jun. 2022.
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Articles