arXiv subjects

Sudev Naduvath

Publications and source records attributed to Sudev Naduvath.

At least 19 recordsLinked to original sources

arXiv2025

$s$-Shunt Intersection Graph of a Graph

The intersection graph of a family of sets $\{S_{1},S_{2},\ldots,S_{n}\}$ is a graph whose vertex set is $\{S_{1},S_{2},\ldots,S_{n}\}$ and two distinct vertices are adjacent if the intersection of the corresponding sets is non-empty. Different types of intersection graphs have been studied depending on the nature of sets taken as the vertex set. A study on a particular type of intersection graph called $s$-shunt intersection graph, generated from the $s$-arcs of a given graph is initiated in this paper.

math.CO↗

arXiv2024

Equitable Dominator Coloring of Graphs

This paper introduces a new variant of domination-related coloring of graphs, which is a combination of their dominator coloring and equitable coloring called the equitable dominator coloring. An equitable coloring is a proper coloring in which the number of vertices in each color class differs by at most one. In this newly introduced coloring, an additional condition of equitability is added to the existing concept of dominator coloring. The minimum number of colors used in this coloring is called the equitable dominator chromatic number, denoted by $\chi_{ed}(G)$. The concept of equitable dominator coloring is explored for basic graph classes. The equitable dominator chromatic number is obtained for the same, and some observations on the bounds are made in this paper.

math.CO↗

arXiv2024

Equitable Dominator Coloring of Line Graphs of Some Graphs

A proper vertex coloring of the graph G such that each vertex dominates at least one color class and the cardinalities of the color classes differ by at most 1 is called an equitable dominator coloring of G. The minimum number of colors used in this coloring is called the equitable dominator chromatic number (EDCN), represented by \chi_{ed}(G). This article explores the concept of equitable dominator coloring for the line graph L(G) of some graph classes.

math.CO↗

arXiv2020

On Certain Topological Indices of Signed Graphs

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical difference of degrees of its end vertices and the irregularity of $G$ is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs.

math.GM↗

arXiv2019

On Derivative Euler Phi Function Set-Graphs

In this paper, we study some graph theoretical properties of two derivative Euler Phi function set-graphs. For the Euler Phi function $ϕ(n)$, $n\in \mathbb{N}$, the set $S_ϕ(n) =\{i:\gcd(i,n)=1, 1\leq i \leq n\}$ and the vertex set is $\{v_i:i\in S_ϕ(n)\}$. Two graphs $G_d(S_ϕ(n))$ and $G_p(S_ϕ(n))$, defined with respect to divisibility adjacency and relatively prime adjacency conditions, are studied.

math.GM↗

arXiv2018

Chromatic Schultz Polynomial of Certain Graphs

A topological index of a graph $G$ is a real number which is preserved under isomorphism. Extensive studies on certain polynomials related to these topological indices have also been done recently. In a similar way, chromatic versions of certain topological indices and the related polynomials have also been discussed in the recent literature. In this paper, the chromatic version of the Schultz polynomial is introduced and determined this polynomial for certain fundamental graph classes.

math.GM↗

arXiv2018

Chromatic Topological Indices of Certain Cycle Related Graphs

Topological indices are real numbers invariant under graph isomorphisms. Chromatic analogue of topological indices has been introduced recently in literature in 2017. Mainly, chromatic versions of Zagreb indices are studied lately. This paper discusses the notion of chromatic topological and irregularity indices of certain cycle related graphs.

math.GM↗

arXiv2018

On $J$-Colouring of Chithra Graphs

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.

math.GM↗

arXiv2018

$δ^{(k)}$-Colouring of Cycle Related Graphs

With respect to a proper colouring of a graph $G$, we know that $δ(G) \leq χ(G) \leq Δ(G)+1$. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of each of $k$, $1\leq k < χ(G)$ technology types together with the minimum adjacency between similar technology types. In an improper colouring an edge $uv$ such that $c(u)=c(v)$ is called a bad edge. In this paper, we introduce the notion of $δ^{(k)}$-colouring which is a near proper colouring of $G$ with exactly $1\leq k < χ(G)$ distinct colours which minimizes the number of bad edges.

math.GM↗

arXiv2018

A Note on $J$-Colouring of Jahangir Graphs

In this paper, we discuss $J$-colouring of the family of Jahangir graphs. Note that the family of Jahangir graphs is a wide-ranging family of graphs which by a generalised definition includes wheel graphs. We characterise the subset of Jahangir graphs which admit a $J$-colouring.

math.GM↗

arXiv2018

Some New Results on Proper Colouring of Edge-set Graphs

In this paper, we present a foundation study for proper colouring of edge-set graphs. The authors consider that a detailed study of the colouring of edge-set graphs corresponding to the family of paths is best suitable for such foundation study. The main result is deriving the chromatic number of the edge-set graph of a path, $P_{n+1}$, $n \geq 1$. It is also shown that edge-set graphs for paths are perfect graphs.

math.GM↗

arXiv2018

On Chromatic Core Subgraph of Simple Graphs

If distinct colours represent distinct technology types that are placed at the vertices of a simple graph in accordance to a minimum proper colouring, a disaster recovery strategy could rely on an answer to the question: "What is the maximum destruction, if any, the graph (a network) can undergo while ensuring that at least one of each technology type remain, in accordance to a minimum proper colouring of the remaining induced subgraph." In this paper, we introduce the notion of a chromatic core subgraph $H$ of a given simple graph $G$ in answer to the stated problem. Since for any subgraph $H$ of $G$ it holds that $χ(H) \leq χ(G)$, the problem is well defined.

math.GM↗

arXiv2018

Some Properties of Fibonacci-Sum Set-Graphs

In this paper we study some properties of Fibonacci-sum set-graphs. The aforesaid graphs are an extension of the notion of Fibonacci-sum graphs to the notion of set-graphs. The colouring of Fibonacci-sum graphs is also discussed. A number of challenging research problems are posed in the conclusion.

math.GM↗

arXiv2017

On the Rainbow Neighbourhood Number of Set-Graphs

In this paper, we present results for the rainbow neighbourhood numbers of set-graphs. It is also shown that set-graphs are perfect graphs. The intuitive colouring dilemma in respect of the rainbow neighbourhood convention is clarified as well. Finally, the new notion of the maximax independence, maximum proper colouring of a graph and a new graph parameter called the $i$-max number of $G$ are introduced as a new research direction.

math.GM↗

arXiv2017

On the Fading Number of a Graph

The closed neighbourhood $N[v]$ of a vertex $v$ of a graph $G$, consisting of at least one vertex from all colour classes with respect to a proper colouring of $G$, is called a rainbow neighbourhood in $G$. The minimum number of vertices and the maximum number of vertices which yield rainbow neighbourhoods with respect to a chromatic colouring of $G$ are called the minimum and maximum rainbow neighbourhood numbers, denoted by $r^-_χ(G)$, $r^+_χ(G)$ respectively. In this paper, by a colour, we mean a solid colour and by a transparent colour, we mean the fading of a solid colour. The fading numbers of a graph $G$, denoted by $f^-(G)$, $f^+(G)$ respectively, are the maximum number of vertices for which the colour may fade to transparent without a decrease in $r^-_χ(G)$ and $r^+_χ(G)$ respectively.

math.GM↗

arXiv2017

Reflection on rainbow neighbourhood numbers of graphs

A rainbow neighbourhood of a graph $G$ with respect to a proper colouring $\C$ of $G$ is the closed neighbourhood $N[v]$ of a vertex $v$ in $G$ such that $N[v]$ consists of vertices from all colour classes in $G$ with respect to $\C$. The number of vertices in $G$ which yield a rainbow neighbourhood of $G$ is called its rainbow neighbourhood number. In this paper, we show that all results known so far about the rainbow neighbourhood number of a graph $G$ implicitly refer to a minimum number of vertices which yield rainbow neighbourhoods in respect of the minimum proper colouring where the colours are allocated in accordance with the rainbow neighbourhood convention. Relaxing the aforesaid convention allows for determining a maximum rainbow neighbourhood number of a graph $G$. We also establish the fact that the minimum and maximum rainbow neighbourhood numbers are respectively, unique and therefore a constant for a given graph.

math.GM↗

arXiv2017

On J-Colorability of Certain Derived Graph Classes

A vertex $v$ of a given graph $G$ is said to be in a rainbow neighbourhood of $G$, with respect to a proper coloring $C$ of $G$, if the closed neighbourhood $N[v]$ of the vertex $v$ consists of at least one vertex from every colour class of $G$ with respect to $C$. A maximal proper colouring of a graph $G$ is a $J$-colouring of $G$ if and only if every vertex of G belongs to a rainbow neighbourhood of $G$. In this paper, we study certain parameters related to $J$-colouring of certain Mycielski type graphs.

math.GM↗

arXiv2017

Rainbow Neighbourhood Equate Number of Graphs

In this paper, a new invariant of a graph namely, the rainbow neighbourhood equate number of a graph $G$ denoted by $ren(G)$ is introduced. It is defined to be the minimum number of vertices whose removal results in a subgraph that admits a $J$-colouring. The new notions of chromatic degree of a vertex $d_χ(v)$, the maximum and minimum chromatic degrees of $G$ denoted, $Δ_χ(G)$ and $δ_χ(G)$ respectively, are also introduced. The chromatic diameter of $G$ denoted, $d(G,χ)$ is introduced as well. The study of $ren(G)$ appears to be very complex for graphs in general so for now, only introductory results will be presented. Finally, the concept of a chromatic degree sequence is proposed as a new research direction.

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