arXiv subjects
Publications and source records attributed to Sudev Naduvath.
In this paper, a new type of colouring called Johan colouring is introduced. This colouring concept is motivated by the newly introduced invariant called the rainbow neighbourhood number of a graph. The study ponders on maximal colouring opposed to minimum colouring. An upper bound for a connected graph is presented and a number of explicit results are presented for cycles, complete graphs, wheel graphs and for a complete $l$-partite graph.
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.
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.
For a positive integer $n$, let $\mZ$ be the set of all non-negative integers modulo $n$ and $\sP(\mZ)$ be its power set. A modular sumset valuation or a modular sumset labeling of a given graph $G$ is an injective function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ defined by $f^+ (uv) = f(u)+ f(v)$. A sumset indexer of a graph $G$ is an injective sumset valued function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ is also injective. In this paper, some properties and characteristics of this type of modular sumset labeling of graphs are being studied.
In this paper, we introduce the notion of a finite non-simple directed graph, called an ornated graph and initiate a study on ornated graphs. An ornated graph is a directed graph on $n$ vertices, denoted by $O_n(s_l)$, whose vertices are consecutively labeled clockwise on the circumference of a circle and constructed from an ordered string $s_l$ joining them in such a way that for an odd indexed entry $a_t$ of the string, a tail $v_i$ has clockwise heads $v_j$ if and only if $(i+a_t) \ge j$ and for an even indexed entry $a_s$ of the string a tail $v_i$ has anticlockwise heads $v_j$ if and only if $(i-a_s) \le j$. The collection of the ornated graphs having this property is called the family of ornated graphs. Some interesting results are also presented in this paper on certain types of ornated graphs.
The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) = \frac{1}{2}\sum\limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total irregularity of simple undirected graphs with at least one cut edge is introduced. We also introduce the concept of an edge-joint between two simple undirected graphs. We also introduce the concept of total irregularity in respect of in-degree and out-degree in simple directed graphs. These invariants are called total in-irregularity and total out-irregularity respectively. In this paper, we initiate a study on these parameters of given simple undirected graphs and simple digraphs.