The graph of a base power b, associated to a positive integer number

https://doi.org/10.5281/zenodo.5728168

• Daniel Brito Department of Mathematics, University of Oriente. https://orcid.org/0000-0002-4925-2976
• Oscar Castro Department of Mathematics, University of Oriente https://orcid.org/0000-0002-3882-7033
• Lope Marı́n Department of Mathematics, University of Oriente. https://orcid.org/0000-0002-4137-0572

Keywords: Hamiltonian Cycle, Hamilton-connectivity, Pancyclicity

Abstract

Many concepts of Number Theory were used in Graph Theory and several types of graphs have been introduced. We introduced the graph of a base power $b \in \mathbb{Z}^{+}-\{1\}$, associated to a positive integer number $n \in \mathbb{Z}^{+} $, denoted for $GP_{b}(n)$, with set of vertices $V=\{x\}_{x=1}^{n}$ and with set of edges:

$$ E =\{ \{x, y\} \in 2^{\,V}: \exists r \in \mathbb{Z}^{+}\cup\{ 0 \} \mbox{, such that } | y \,-\, x | = b^{\,r} \} \mbox{,} $$

and we study some of its properties, in special for case $b=2$.

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