For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

Preface: graph colorings

This paper describes the bibliometric characteristics of 1,319 articles published in the journal MATCH Communications in Mathematical and in Computer Chemistry, in the period 1995-2014. These articles have been analysed using data from the Web of Science Core Collection and demonstrate the contribution of the journal not only to mathematical and computer chemistry but also to science more generally.

A Bibliometric Profile of MATCH Communications in Mathematical and in Computer Chemistry

The notion of the b-chromatic number of a graph attracted much research interests and recently a new concept, namely the b-chromatic sum of a graph, denoted by φ′(G), has also been introduced. Motivated by the studies on b-chromatic sum of graphs, in this paper we introduce certain new parameters such as χ-chromatic sum, χ+-chromatic sum, b+-chromatic sum, π-chromatic sum and π+-chromatic sum of graphs. We also discuss certain results on these parameters for a selection of standard graphs.

Generalised colouring sums of graphs

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 $v_i$, $i n$. 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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A Study on Linear Jaco Graphs

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 rχ(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.

/pdf/rainbow-neighbourhood-number-of-graphs-55ywc5s1d3.pdf

Rainbow neighbourhood number of graphs

Let G be a finite and simple undirected connected graph of order n ≥ 1 and let φ be a proper vertex coloring of G. Denote φ : vi↦cj simply, c(vi) = cj. In this paper, we introduce a variation of the well-known Zagreb indices by utilizing 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.

On Chromatic Zagreb Indices of Certain Graphs

Coloring the vertices of a graph G according to certain conditions is a random experiment and a discrete random variable X is defined as the number of vertices having a particular color in the given type of coloring of G and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph G is a proper coloring &#x1d49e; of G which an assignment of colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

Johan Kok

Papers

Generalised colouring sums of graphs

A Study on Linear Jaco Graphs

Rainbow neighbourhood number of graphs

On Chromatic Zagreb Indices of Certain Graphs

On certain parameters of equitable coloring of graphs