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Johan Kok

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Papers by Johan Kok

On Certain Colouring Parameters of Mycielski Graphs of Some Graphs

arXiv (Cornell University), Aug 4, 2017

Colouring the vertices of a graph G according to certain conditions can be considered as a random... more Colouring the vertices of a graph G according to certain conditions can be considered as a random experiment and a discrete random variable X can be defined as the number of vertices having a particular colour in the proper colouring of G. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph colouring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the colouring parameters of the Mycielskian of certain standard graphs.

Chromatic Zagreb indices for graphical embodiment of colour clusters

arXiv (Cornell University), Oct 31, 2016

For a colour cluster C = (C 1 , C 2 , C 3 , . . . , C ), where C i is a colour class such that |C... more For a colour cluster C = (C 1 , C 2 , C 3 , . . . , C ), where C i is a colour class such that |C i | = r i , a positive integer, we investigate two types of simple connected graph structures G C 1 , G C 2 which represent graphical embodiments of the colour cluster such that the chromatic numbers 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 C 1 , G C 2 .

Jaco-Type Graphs and Black Energy Dissipation

arXiv (Cornell University), Jul 2, 2016

In this paper, we introduce the notion of an energy graph as a simple, directed and vertex labele... more In this paper, we introduce the notion of an energy graph as a simple, directed and vertex labeled graph such that the arcs (u i , u j ) / ∈ A(G) if i > j for all distinct pairs u i , u j and at least one vertex 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.

On $J$-Colouring of Chithra Graphs

arXiv (Cornell University), Aug 26, 2018

The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size ... more The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one. Chithra graphs serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within various biological networks found in living systems. In this paper, we discuss recently introduced J-colouring of the family of Chithra graphs.

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

HAL (Le Centre pour la Communication Scientifique Directe), 2019

The concepts of linear Jaco graphs and Jaco-type graphs have been introduced as certain types of ... more 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.

A study on the injective coloring parameters of certain graphs

Discrete Mathematics, Algorithms and Applications, Aug 1, 2016

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be consi... more Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

An Essay on Compônentă Analysis of Graphs

arXiv (Cornell University), May 8, 2017

Rainbow Neighbourhoods of Graphs

arXiv (Cornell University), Mar 3, 2017

In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter... more In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex of each colour in the chromatic colouring of a graph is called a rainbow neighbourhood. The number of rainbow neighbourhoods in a graph G is called the rainbow neighbourhood number of G, denoted by r χ (G). We also introduce the concepts of an expanded line graph of a graph G and a v-clique of v ∈ V (G). With the help of these new concepts, we also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph G.

Propagating Graphs and Black Energy Dissipation

arXiv (Cornell University), Jul 2, 2016

In this paper, we introduce the notion of a propagating graph as a simple, directed and vertex la... more In this paper, we introduce the notion of a propagating graph as a simple, directed and vertex labeled graph G such that the arcs 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 points 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 Jacotype graphs are considered for the application of the new concepts.

A Study on Sprout Graphs

arXiv (Cornell University), Mar 27, 2014

Sprout graphs are finite directed graphs matured over a finite subset of the non-negative time li... more Sprout graphs are finite directed graphs matured over a finite subset of the non-negative time line. A simple undirected connected graph on at least two vertices is required to construct an infant graph to mature from. The maxi-max arc-weight principle and the maxi-min arc-weight principle are introduced in this paper to determine the maximum and minimum maturity weight of a sprout graph. These principles demand more mathematical debates for logical closure. Since complete graphs, paths, stars and possibly cycles form part of the skeleton of all graphs, the introduction of results for these family of sprout graphs is expected to lay a good research foundation.

A Study on Primitive Holes of Certain Graphs

HAL (Le Centre pour la Communication Scientifique Directe), 2014

On Chromatic Zagreb Indices of Certain Graphs

arXiv (Cornell University), May 2, 2016

Let G be a finite and simple undirected connected graph of order n ≥ 1 and let ϕ be a proper vert... more Let G be a finite and simple undirected connected graph of order n ≥ 1 and let ϕ be a proper vertex colouring of G. Denote ϕ : v i → c j simply, c(v i ) = c j . In this paper, we introduce a variation of the well-known Zagreb indices by utilising the parameter c(v) instead of the invariant d(v) for all vertices of G. The new indices are called chromatic Zagreb indices. We study these new indices for certain classes of graphs and introduce the notion of chromatically stable graphs.

The $b$-Chromatic Number of Certain Graphs and Digraphs

arXiv (Cornell University), Nov 2, 2015

The b-chromatic number of a graph is the maximum number k of colors that can be used to color the... more The b-chromatic number of a graph is the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i ≤ k, has at least one representant x i adjacent to a vertex of every color j, 1 ≤ j = i ≤ k. The studies on b-chromatic number attracted much interest since its introduction. In this paper, we discuss the b-chromatic number of certain classes of graphs and digraphs. The notion of a new general family of graphs called the Chithra graphs of a graph G are also introduced in this study and the corresponding b-chromatic numbers are studied.

$Research paper thumbnail of An Essay on Comp\^onent\u{a} Analysis of Graphs$

An Essay on Comp\^onent\u{a} Analysis of Graphs

arXiv (Cornell University), May 1, 2017

In most studies related to colouring of graphs and perhaps in the study of other invariants and v... more 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 compônentȃ analysis, these restrictions are relaxed. In particular, this essay focuses on compônentȃ analysis in respect of the recently introduced J-colouring. The concept of J c -rainbow connectivity is also introduced in this paper.

$Research paper thumbnail of Contemplating some invariants of the Jaco Graph, $J_n(1), n \in \Bbb N$$

Contemplating some invariants of the Jaco Graph, $J_n(1), n \in \Bbb N$

arXiv (Cornell University), Oct 30, 2014

introduced Jaco Graphs (order 1 ). In this essay we present a recursive formula to determine the ... more introduced Jaco Graphs (order 1 ). In this essay we present a recursive formula to determine the independence number α(J n (1)) = |I| with, We also prove that for the Jaco Graph, J n (1), n ∈ N with the prime Jaconian vertex v i the chromatic number, χ(J n (1)) is given by: We further our exploration in respect of domination numbers, bondage numbers and declare the concept of the murtage number 2 of a simple connected graph G, denoted m(G). We conclude by proving that for any Jaco Graph J n (1), n ∈ N we have that 0 ≤ m(J n (1)) ≤ 3.

Some Results on the $b$-Colouring Parameters of Graphs

arXiv: General Mathematics, 2017

A vertex colouring of a given graph $G$ can be considered as a random experiment. A discrete rand... more A vertex colouring of a given graph $G$ can be considered as a random experiment. A discrete random variable $X$, corresponding to this random experiment, can be defined as the colour of a randomly chosen vertex of $G$ and a probability mass function for this random variable can be defined accordingly. In this paper, we study the concepts of mean and variance corresponding to the $b$-colouring of $G$ and hence determine the values of these parameters for a number of standard graphs.

A study on the curling number of graph classes

arXiv: General Mathematics, 2015

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ ... more 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.

Competition Graphs of Jaco Graphs and the Introduction of the Grog Number of a Simple Connected Graph

arXiv: Combinatorics, 2015

Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be ... more Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be a non-empty subset of $V(G^\rightarrow)$ and denote the undirected subgraph induced by $V^*$ by, $\langle V^* \rangle.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1), n \in \Bbb N, n \geq 5,$ denoted by $C(J_n(1))$ is given by:\\ \\ $C(J_n(1)) = \langle V^* \rangle_{V^* = \{v_i|3 \leq i \leq n-1\}} - \{v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2\} \cup \{v_1, v_2, v_n\}.$\\ \\ Further to the above, the concept of the \emph{grog number} $g(G^\rightarrow)$ of a simple connected directed graph $G^\rightarrow$ on $n \geq 2$ vertices as well as the general \emph{grog number} of the underlying graph $G$, will be introduced. The \emph{grog number} measures the efficiency of an \emph{optimal predator-prey strategy} if the simple directed graph models an ecological predator-prey web.\\ \\ We also pose four open problems for exploratory research.

Some New Graph Coloring Problems

Curling Sub-sequences and Curling Number of Certain Graphs

arXiv: Combinatorics, 2015

In this paper, we introduce the concept of curling subsequence of simple, finite and connected gr... more In this paper, we introduce the concept of curling subsequence of simple, finite and connected graphs. A curling subsequence is a maximal subsequence $C$ of the degree sequence of a simple connected graph $G$ for which the curling number $cn(G)$ corresponds to the curling number of the degree sequence per se and hence we call it the curling number of the graph $G$. A maximal degree subsequence with equal entries is called an identity subsequence. The number of identity curling subsequences in a simple connected graph $G$ is denoted $ic(G).$ We show that the curling number conjecture holds for the degree sequence of a simple connected graph $G$ on $n \geq 1$ vertices. We also introduce the notion of the compound curling number of a simple connected graph $G$ and then initiate a study on the curling number of Jaco graphs.