A Study on Linear Jaco Graphs

Authors

  • Johan Kok Tshwane Metropolitan Police Department, City of Tshwane
  • Susanth C Department of Mathematics, Vidya Academy of Science and Technology, Thalakkottukara, Thrissur-680501
  • Sunny Joseph Kalayathankal Department of Mathematics, Kuriakose Elias College), Mannanam, Kottayam- 686561, Kerala

DOI:

https://doi.org/10.26713/jims.v7i2.291

Keywords:

Linear function Jaco graph, Hope graph, Directed graph, Jaconian vertex, Jaconian set

Abstract

We introduce the concept of a family of nite directed graphs (positive integer order, $f(x) =mx+c$; $x, m \in \mathbb{N}$ and $c \in \mathbb{N}_0$) which are directed graphs derived from an innite directed graph called the $f(x)$-root digraph. The $f(x)$-root digraph has four fundamental properties which are; $V (J_\infty(f(x))) = \{v_i : i \in \mathbb{N}\}$ and, if $v_j$ is the head of an arc then the tail is always a vertex $v_i$, $i < j$ and, if $v_k$ for smallest $k \mathbb{N}$ is a tail vertex then all vertices $v_\ell$, $k < \ell < j$ are tails of arcs to $v_j$ and finally, the degree of vertex $v_k$ is $d(v_k) = mk + c$. The family of nite directed graphs are those limited to $n \in \mathbb{N}$ vertices by lobbing o all vertices (and corresponding arcs) $v_t$, $t > n$. Hence, trivially we have $d(v_i) mi + c$ for $i \in \mathbb{N}$. It is meant to be an introductory paper to encourage further research.

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Author Biography

Johan Kok, Tshwane Metropolitan Police Department, City of Tshwane

Director: Licensing Services, City of Tshwane

References

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J. Kok, P. Fisher, B. Wilkens, M. Mabula and V. Mukungunugwa, Characteristics of Finite Jaco Graphs, $J_n(1)$, $n in mathbb{N}$, arXiv: 1404.0484v1 [math.CO], 2 April 2014.

J. Kok, P. Fisher, B. Wilkens, M. Mabula and V. Mukungunugwa, Characteristics of Jaco Graphs, $J_infty(a)$, $ a in mathbb{N}$, arXiv: 1404.1714v1 [math.CO], 7 April 2014.

D.B. West, Introduction to Graph Theory, Pearson Education Incorporated, 2001.

E. Zeckendorf, Repreesentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bulletin de la Societe Royale des Sciences de Liege 41 (1972), 179-182.

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Published

2015-11-30
CITATION

How to Cite

Kok, J., C, S., & Kalayathankal, S. J. (2015). A Study on Linear Jaco Graphs. Journal of Informatics and Mathematical Sciences, 7(2), 69–80. https://doi.org/10.26713/jims.v7i2.291

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Section

Research Articles