arXiv subjects

Johan Kok

Publications and source records attributed to Johan Kok.

At least 37 records · Page 2Linked to original sources

arXiv2017

An Essay on Compônentă Analysis of Graphs

In most studies related to colouring of graphs and perhaps in the study of other invariants and variants of graphs, the restrictions of non-triviality and connectedness are placed upon graphs. For the introduction to compônentă analysis, these restrictions are relaxed. In particular, this essay focuses on compônentă analysis in respect of the recently introduced $J$-colouring. The concept of $J^c$-rainbow connectivity is also introduced in this paper.

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arXiv2017

Johan Colouring of Graph Operations

A vertex $v$ of a given graph is said to be in a rainbow neighbourhood of $G$ if every colour class of $G$ consists of at least one vertex from the closed neighbourhood $N[v]$. A maximal proper colouring of a graph $G$ is a Johan colouring if and only if every vertex of $G$ belongs to a rainbow neighbourhood of $G$. In general all graphs need not have a Johan colouring, even though they admit a chromatic colouring. In this paper, we characterise graphs which admit a Johan colouring. We also discuss some preliminary results in respect of certain graph operations which admit a Johan colouring under certain conditions.

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arXiv2017

Rainbow Neighbourhoods of Graphs

In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely, the rainbow neighbourhood number of a graph $G$. We report on preliminary results thereof. We also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph $G$.

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arXiv2017

A note on chromatic blending of colour clusters

For a colour cluster $\C =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\dots,\mathcal{C}_\ell)$, $\mathcal{C}_i$ is a colour class, and $|\mathcal{C}_i|=r_i \geq 1$, we investigate a simple connected graph structure $G^{\C}$, which represents a graphical embodiment of the colour cluster such that the chromatic number $χ(G^{\C})= \ell,$ and the number of edges is a maximum, denoted $\varepsilon^+(G^{\C})$. We also extend the study by inducing new colour clusters recursively by blending the colours of all pairs of adjacent vertices. Recursion repeats until a maximal homogeneous blend between all $\ell$ colours is obtained. This is called total chromatic blending. Total chromatic blending models for example, total genetic, chemical, cultural or social orderliness integration.

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arXiv2016

Chromatic Zagreb indices for graphical embodiment of colour clusters

For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, where $\mathcal{C}_i$ is a colour class such that $|\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $χ(G^{\mathbb{C}}_1)=χ(G^{\mathbb{C}}_2)=\ell$ and $\min\{\varepsilon(G^{\mathbb{C}}_1)\}=\min\{\varepsilon(G^{\mathbb{C}}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$. Therefore, the problem is the edge-minimality inverse to finding the chromatic number of a given simple connected graph. In this paper, we also discuss the chromatic Zagreb indices corresponding to $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$.

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arXiv2016

Jaco-Type Graphs and Black Energy Dissipation

In this paper, we introduce the notion of an energy graph as a simple, directed and vertex labeled graph $G$ such that the arcs $(v_i, v_j) \notin A(G)$ if $i > j$ for all distinct pairs $v_i,v_j$ and at least one vertex $v_k$ exists such that $d^-(v_k)=0$. Initially, equal amount of potential energy is allocated to certain vertices. Then, at a point of time these vertices transform the potential energy into kinetic energy and initiate transmission to head vertices. Upon reaching a head vertex, perfect elastic collisions with atomic particles take place and propagate energy further. Propagation rules apply which result in energy dissipation. This dissipated energy is called black energy. The notion of the black arc number of a graph is also introduced in this paper. Mainly Jaco-type graphs are considered for the application of the new concepts.

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arXiv2016

A Study on Clique Invariants of Jaco-type Graphs

The first study related to this paper was on the notion of primitive holes. This paper reports on research in respect of clique parameters and related properties thereof within Jaco-type graphs.

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arXiv2016

A Study on the Quadratic Jaco Graph

In this paper we introduce the quadratic Jaco graph. The characteristics, properties and some graph invariants of quadratic Jaco graphs are discussed. The observation that quadratic Jaco graphs are well-defined in respect of complete graphs and bridges led to the notion of braided complete graphs.

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arXiv2016

Some New Results on the Curling Number of Graphs

Let $S=S_1S_2S_3\ldots S_n$ be a finite string. Write $S$ in the form $XYY\ldots Y=XY^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest value of this integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. Let the degree sequence of the graph $G$ be written as a string of identity curling subsequences say, $X^{k_1}_1\circ X^{k_2}_2\circ X^{k_3}_3 \ldots \circ X^{k_l}_l$. The compound curling number of $G$, denoted $cn^c(G)$ is defined to be, $cn^n(G) = \prod\limits^{l}_{i=1}k_i$. In this paper, we discuss the curling number and compound curling number of certain products of graphs.

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arXiv2016

On the Vertex In-Degrees of Certain Jaco-Type Graphs

The concepts of linear Jaco graphs and Jaco-type graphs have been introduced as certain types of directed graphs with specifically defined adjacency conditions. The distinct difference between a pure Jaco graph and a Jaco-type graph is that for a pure Jaco graph, the total vertex degree $d(v)$ is well-defined, while for a Jaco-type graph the vertex out-degree $d^+(v)$ is well-defined. Hence, in the case of pure Jaco graphs a challenge is to determine $d^-(v)$ and $d^+(v)$ respectively and for Jaco-type graphs a challenge is to determine $d^-(v)$. In this paper, the vertex in-degrees for Fibonaccian and modular Jaco-type graphs are determined.

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arXiv2016

A study on the curling number of graph classes

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ (which may possibly be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.

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arXiv2016

Certain Chromatic Sums of Some Cycle Related Graph Classes

Let $\mathcal{C} = \{c_1,c_2, c_3, \ldots,c_k\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $θ(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring sum of a given graph $G$ with respect to the colouring $\cC$, denoted by $ω_{\cC}(G)$, is defined to be $ω(\cC) = \sum\limits_{i=1}^{k}i\,θ(c_i)$. The colouring sums such as $χ$-chromatic sum, $χ^+$-chromatic sum, $b$-chromatic sum, $b^+$-chromatic sum etc. are some of these types of colouring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle related graphs.

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arXiv2016

On Chromatic Zagreb Indices of Certain Graphs

In this paper we introduce a variation of the well-known Zagreb indices by considering a proper vertex colouring of a graph $G$. The chromatic Zagreb indices are defined in terms of the parameter $c(v), v \in V(G)$ instead of the invariant $d_G(v)$. The notion of chromatically stable graphs is also introduced.

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arXiv2016

A Note on the Tattoo Index of Graphs

Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The tattooing of a simple connected directed graph $D$ is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of $D$ is a corresponding derivative of brush numbers in it. In this paper, we introduce a new concept, called the tattoo index of a given graph $G$, which is an efficiency index related to the tattooing sequence and we establish some introductory results on this parameter.

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arXiv2016

Tattooing and the Tattoo Number of Graphs

Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The minimum number of brushes required for cleaning the network $D$ is called its brush number. The tattooing of a simple connected directed graph $D$ is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of $D$ is a corresponding derivative of brush numbers in it. Tattooing along an out-arc of a vertex $v$ may proceed if a minimum set of colour-brushes is allocated (primary colours) or combined with those which have arrived (including colour blends) together with mutation of permissible new colour blends, has cardinality greater than or equal to $d^+_G(v)$.

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arXiv2016

Coloring Sums of Extensions of Certain Graphs

Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $χ(G).$ In this paper the concepts of $χ$'-chromatic sum and $χ^+$-chromatic sum are introduced. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. We further the concepts of $χ$'-chromatic sum and $χ^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.

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arXiv2016

Total Irregularity and $f_t$-Irregularity of Linear Jaco Graphs$

Total irregularity of a simple undirected graph $G$ is defined to be $irr_t(G) = \frac{1}{2}\sum\limits_{u, v \in V(G)}|d(u) - d(v)|$. See Abdo and Dimitrov [2]. We allocate the \emph{Fibonacci weight,} $f_i$ to a vertex $v_j$ of a simple connected graph, if and only if $d(v_j) = i$ and define the \emph{total fibonaccian irregularity} or $f_t-irregularity$ denoted $firr_t(G)$ for brevity, as: $firr_t(G) = \sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}|f_i - f_j|.$ The concept of an \emph{edge-joint} is also introduced to be the simple undirected graph obtained from two simple undirected graphs $G$ and $H$ by linking the edge $vu_{v \in V(G), u \in V(H)}$. This paper presents results for the undirected underlying graphs of Jaco Graphs, $J_n(x)$. Finally we pose an open problem with regards to $firr_t^\pm(G).$

math.CO↗

arXiv2016

On New Thue Colouring Concepts of Certain Graphs

The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in $G$ is non-repetitive. The change in the Thue number, $π(G)$, as edges are iteratively removed from a graph $G$ is studied. The notion of the $τ$-index denoted, $τ(G)$, of a graph $G$ is introduced as well. $τ(G)$ serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.

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