Leomarich F. Casinillo

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 1, Pages 8—20

DOI: 10.7546/nntdm.2020.26.1.8-20

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## Details

### Authors and affiliations

Leomarich F. Casinillo

*Department of Mathematics and Physics, Visayas State University
Visca, Baybay City, Leyte, Philippines
*

### Abstract

Let {P_{n}}_{n=1}^{∞} be a sequence of paths. The odd repetition sequence denoted by {𝜌_{𝑘}^{𝑜}:𝑘∈ℕ} is a sequence of natural numbers in which odd numbers are repeated once and defined by {𝜌_{𝑘}^{𝑜}}={1,1,2,3,3,4,5,5,…}={𝑖(𝑃_{𝑛})} where 𝑛=2𝑘−1. The even repetition sequence denoted by {𝜌_{𝑘}^{𝑒}:𝑘∈ℕ} is a sequence of natural numbers, in which even numbers are repeated once and defined by {𝜌_{𝑘}^{𝑒}}={1,2,2,3,4,4,5,6,6,…}={𝑖(𝑃_{𝑛})}, where 𝑛 = 2𝑘. In this paper, the explicit formula that shows the values of the element of two sequences {𝜌_{𝑘}^{𝑜}}} and {𝜌_{𝑘}^{𝑒}} that depends on the subscript 𝑘 were constructed. Also, the formula that relates the partial sum of the elements of the said sequences, which depends on the subscript 𝑘 and order of the sequence of paths, were established. Further, the independent domination number of the triangular grid graph 𝑇_{𝑚} = (𝑉(𝑇_{𝑚}), 𝐸(𝑇_{𝑚})) will be determined using the said sequences and the two sequences will be evaluated in relation to the Fibonacci sequence {𝐹_{𝑛}} along with the order of the path.

### Keywords

- Odd repetition sequence
- Even repetition sequence
- Independent domination number
- Fibonacci numbers
- Triangular grid graph

### 2010 Mathematics Subject Classification

- 05C69

### References

- Canoy, Jr. S. R., & Garces, I. J. L. (2002). Convex sets under some graph operations. Graphs and Combinatorics, 18, 787–793.
- Casinillo, L. F. (2018). A note on Fibonacci and Lucas number of domination in path. Electronic Journal of Graph Theory and Applications, 6(2), 317–325.
- Casinillo, L. F., Lagumbay, E. T. & Abad, H. R. F. (2017). A note on connected interior domination in join and corona of two graphs. IOSR Journal of Mathematics, 13(2), 66–69.
- Chartrand, G. & Zhang, P. (2012). A First Course in Graph Theory. Dover Publication Inc., New York.
- Cockayne, E. J., & Hedetniemi, S. T. (1977). Towards a theory of domination in graph. Networks Advanced Topics, 7, 247–261.
- Dorfling, M., & Henning, M. A. (2006). A note on power domination in grid graphs. Discrete Applied Mathematics, 154, 1023–1027.
- Haynes, T. W., Hedetniemi, S. T., & Slater, P. J. (1998). Fundamentals of Domination in Graphs. Marcel Dekker. New York.
- Koshy, T. (2001). Fibonacci and Lucas Numbers with Application. Wiley-Interscience, New York.
- Ore, O. (1962). Theory of Graphs. American Mathematical Society Providence, R. I.
- Perderson, A. S., & Vestergaard, P. D. (2005). The number of independent sets in unicyclic graphs. Discrete Applied Mathematics, 152, 246–256.
- Prodinger, H., & Tichy, R. (1982). Fibonacci numbers of graphs. Fibonacci Quarterly, 20(1), 16–21.
- Singh, B. Sisodiya, K., & Ahmad, F. (2014). On the products of k-Fibonacci numbers and k-Lucas numbers. International Journal of Mathematics and Mathematical Science, 21, 1–4.
- Tarr, J. M. (2010). Domination in Graphs. Graduate Theses and Dissertations. Retrieved from https://scholarcommons.usf.edu/etd/1786.
- Vajda, S. (2008). Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Dover Publications Inc., New York.

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## Cite this paper

Casinillo, L. F. (2020). Odd and even repetition sequences of independent domination number. Notes on Number Theory and Discrete Mathematics, 26(1), 8-20, doi: 10.7546/nntdm.2020.26.1.8-20.