One Modulo Three Mean Labeling Of Graphs

Graph labeling was first introduced by Rosa in 1966. Labeling of graphs is an assignment of nonnegative integers to vertices, edges or both according to some specified conditions. Mean labeling of graphs was introduced by Somasundaram and Ponraj in 2003. Subsequently, labelings of graphs were done with geometric mean, harmonic mean etc. In this paper we introduce two concepts called ’Logarithmic mean labeling’, ’Identric mean labeling’ and acquire their mean labelings for some standard graphs.

In graph theory, labeling of vertices and edges according to specific rules offers insight into the structural properties of graphs and their potential applications. This study focuses on a particular type of graph labeling known as Super Lehmer-3 Mean Labeling. A graph is said to admit such a labeling if there exists an injective vertex labeling such that the corresponding edge labels, derived from the Lehmer-3 mean of their incident vertex labels, are also distinct and satisfy certain arithmetic conditions. Graphs that admit such a labeling are termed Super Lehmer-3 Mean Graphs.In this work, we investigate the existence of super Lehmer-3 mean labeling in various families of theta-related graphs. Specifically, we analyze standard theta graphs and their structural variations obtained by subdivisions and attachments of pendant vertices. The results confirm that these modified forms of theta graphs also admit super Lehmer-3 mean labeling. Additionally, we extend our exploration to harmonic mean labeling in selected theta graphs, highlighting the conditions under which this alternative labeling scheme is valid.The findings presented in this paper contribute to the growing body of knowledge in graph labeling theory, offering new characterizations and constructions of labeled graph families. These results may have implications for theoretical studies and applications where such labeling structures are relevant.

A. In this paper, a few technique of coding a message using Graph Message Jumbled (GMJ) method on Star twin graph with subset of super mean labeling and even felicitous labeling is discussed and theorems with illustrations are established.In addition different striking clues on numbering the alphabets, eye-catching pictorial coding and theoretical part for twin graph labeling are provided to increase the secrecy and intricacy for the given message.A coding method for allocating the numbers to the alphabets for encoding the secret messages through graph labelings are developed.

Let G be a simple graph with vertex set V(G) and edge set E(G). Let ⟨ℤ2, +,*⟩ be a field with two elements. A vertex labeling f : V(G) → ℤ2 induces two edge labelings f+: E(G) → ℤ2 such that f+ (xy) = f(x) + f(y), whereas f* : E(G) → ℤ2 such that f* (xy) = f(x) f(y), for each edge xy ∈ E(G). For i ϵ ℤ2, let and . A labeling f of a graph G is said to be friendly if |υf (0) −υf (1)|≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as the vertex labeling f is friendly}. This is a generalization of graph cordiality. The corresponding multiplicative version is called the product-cordial index set, denoted PCI(G), defined as the vertex labeling f is friendly}. In this paper, we investigate the friendly index and product-cordial index sets of a family of cubic graphs known as Möbius-liked graph, MG(n) for even n ≥ 4.

An L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, ..., k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2, 1)-total labeling number λ2T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2T(G) ≤ Δ(G)+2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2T(G) ≤ Δ(G)+2 even in the case of Δ(G) ≤ 4.

All the graphs in this paper are connected graphs. Let $G=(V,E)$ where $V(G)$ is a set of vertex from graph $G$ while $E(G)$ is a set of edge from graph $G$. A bijection function $f: V \rightarrow \{1,2,3,...,\lvert V(G)\rvert\}$ the associated weight of an edge $uv \in E(G)$ under $f$ is $W_f{(uv)}=f(u)+f(v)$. A path $P$ in a vertex-labeled graph $G$ is said to be a rainbow path if for every two edges $uv$, $u'v' \in E(P)$, there is $w_f{(uv)}\neq w_f{u'v'}$. If for every two vertices $u$ and $v$ of $G$, there is a rainbow $u$-$v$ path, then $f$ is called a rainbow antimagic labeling of $G$. A graph $G$ is rainbow antimagic if $G$ has a rainbow antimagic labeling. The minimum number of color needed to make $G$ rainbow connected, called rainbow antimagic connection number, denoted by rac(G). In this paper, we will analyze the rainbow antimagic coloring on comb product of path graph.

All graph in this paper is simple and connected graph where $V(G)$ is vertex set and $E(G)$ is edge set. Let function $f : V(G)\longrightarrow \{0, 2,..., 2k_v\}$ as vertex labeling and a function $f: E(G)\longrightarrow \{1, 2,..., k_e\}$ as edge labeling where $k=max\{2k_v,k_e\}$ for $k_v,k_e$ are natural number. The weight of edge $ u,v\in E(G) $ under $f$ is $w(u)=f(u)+ \Sigma_{uv \in V(G)} f(uv)$. In other words, the function $f$ is called local edge irregular reflexive labeling if every two adjacent edges has distinct weight and weight of a edge is defined as the sum of the labels of edge and the labels of all vertex incident this edge When we assign each edge of $G$ with a color of the edge weight $w(uv)$, thus we say the graph $G$ admits a local edge irregular reflexive coloring. The minimum number of colors produced from local edge irregular reflexive coloring of graph $G$ is reflexive local irregular chromatic number denoted by $\chi_{lrecs}(G).$ Furthermore, the minimum $k$ required such that $\chi_{lrecs}(G)=\chi(G)$ is called a local reflexive edge color strength, denoted by \emph{lrecs}$(G)$. In this paper, we learn about the local edge irregular reflexive coloring and obtain \emph{lrecs}$(G)$ of planar related graphs.

All graph in this paper is simple and connected graph where $V(G)$ is vertex set and $E(G)$ is edge set. Let function $f : V(G)\longrightarrow \{0, 2,..., 2k_v\}$ as vertex labeling and a function $f: E(G)\longrightarrow \{1, 2,..., k_e\}$ as edge labeling where $k=max\{2k_v,k_e\}$ for $k_v,k_e$ are natural number. The weight of vertex $ u,v\in V(G) $ under $f$ is $w(u)=f(u)+ \Sigma_{uv \in E(G)} f(uv)$. In other words, the function $f$ is called local vertex irregular reflexive labeling if every two adjacent vertices has distinct weight and weight of a vertex is defined as the sum of the labels of vertex and the labels of all edges incident this vertex When we assign each vertex of $G$ with a color of the vertex weight $w(uv)$, thus we say the graph G admits a local vertex irregular reflexive coloring. The minimum number of colors produced from local vertex irregular reflexive coloring of graph $G$ is reflexive local irregular chromatic number denoted by $\chi_{lrvs}(G).$ Furthermore, the minimum $k$ required such that $\chi_{lrvs}(G)=\chi(G)$ is called a local reflexive vertex color strength, denoted by \emph{lrvcs}$(G)$. In this paper, we learn about the local vertex irregular reflexive coloring and obtain \emph{lrvcs}$(G)$ of wheel related graphs.

<p>A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. Varieties of graph labeling have been investigated by many authors [2], [3] [5] and they serve as useful models for broad range of applications.</p>

first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessAbstract GraphDraw—A Tool for the Represention of Graphs Using Inherent Symmetry † by Alice Miller 1,*, Dragos Miron 1 and Sharan Maiya 2 1 School of Computing Science, University of Glasgow, Glasgow G12 8RZ, UK 2 School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK * Author to whom correspondence should be addressed. † Presented at Symmetry 2017—The First International Conference on Symmetry, Barcelona, Spain, 16–18 October 2017. Proceedings 2018, 2(1), 86; https://doi.org/10.3390/proceedings2010086 Published: 5 January 2018 (This article belongs to the Proceedings of The First International Conference on Symmetry) Download Download PDF Download PDF with Cover Download XML Versions Notes When drawing small graphs (with up to five or six vertices) we can use their underlying symmetry to represent them in a clear and natural way. For example, to draw a graph G with vertex set V = {0,1,2,3} and edge set E = {(0,1), (1,2), (2,3), (3,0)} it would be natural to use a square, or if V = {0,1,2,3,4} and E = {(0,2), (0,3),(1,3),(1,4),(2,4)} we might use a star to represent G. Even if the vertex labels were permuted, it would not be difficult to uncover the underlying shape. As the number of vertices increases, it becomes impossible to manually choose a suitable and informative representation, i.e., to choose where on the plane to place each of the vertices in order to produce a clear graph, rather than just a confusing mess. Existing graph drawing applications (like yEd) require the user to decide where to place the vertices.In this paper, we present a tool—GraphDraw—which uses the underlying automorphism group of a graph (A(G)) to draw graphs from an initial index array representation of G. Cycles in the generators of A(G) are used to place the vertices in such a way as to exploit the underlying symmetry in the drawing, thus producing a clearer and more intuitive representation. Parameter selection allows the user to choose from a range of representations and to optimise the graphs in terms of edge crossings or total edge length. We give a description of the tool and present a suite of example graphs, illustrating the effect of different parameter selections. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Share and Cite MDPI and ACS Style Miller, A.; Miron, D.; Maiya, S. GraphDraw—A Tool for the Represention of Graphs Using Inherent Symmetry. Proceedings 2018, 2, 86. https://doi.org/10.3390/proceedings2010086 AMA Style Miller A, Miron D, Maiya S. GraphDraw—A Tool for the Represention of Graphs Using Inherent Symmetry. Proceedings. 2018; 2(1):86. https://doi.org/10.3390/proceedings2010086 Chicago/Turabian Style Miller, Alice, Dragos Miron, and Sharan Maiya. 2018. "GraphDraw—A Tool for the Represention of Graphs Using Inherent Symmetry" Proceedings 2, no. 1: 86. https://doi.org/10.3390/proceedings2010086 Find Other Styles Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here. Article Metrics No No Article Access Statistics Multiple requests from the same IP address are counted as one view.

Let G be a graph which has V(G) as a vertex set and E(G) as an edge set. The G graph contain subgraph H that isomorphic with Hj, j = 1, 2, … s, if every e ∈ E(G) include only in one of the edge set of H subgraph (e ∈ E(H)). The total l-labeling is a graph labeling that give label positive integer number until l into vertices and edges. The total H-irregular labeling is a total l-labeling with condition that the sum of vertex labels and edge labels in two distinct subgraphs H 1 and H 2 isomorphic to H is different. We define H-weight as wtφ (H) = ∑ v∈V(H) φ(v) + ∑ e∈E(H) φ(e), for the subgraph H ⊆ G under the total l-labeling (φ). The total H-irregularity strength of G graph (ths(G, H)) is the smallest l value in label of G graph has total H-irregular labeling. We used plane graphs such as triangular ladder graph and windmill graph.