Odd and even repetition sequences of independent domination number

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
Full paper (PDF, 659 Kb)

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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

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