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Mean labelings of graphs
Abstract
We introduce a new type of labeling known as mean labeling. We prove that the following are mean graphs: the path P n , the cycle C n , the complete graph K n for n≤3, the triangular snake and some more special graphs. We also prove that the complete graph K n and the complete bipartite graph K 1,n for n>3 are not mean graphs. From the text: A graph G with p vertices and q edges is called a mean graph if it is possible to label the vertices x∈V with distinct elements f(x) from 0,1,⋯,q in such a way that when each edge e=uv is labelled with (f(u)+f(v))/2 if f(u)+f(v) is even and (f(u)+f(v)+1)/2 if f(u)+f(v) is odd, then the resulting edge labels are distinct. f is called a mean labeling of G.
... At present, there are a lot graph theorist who contributed to the development of labeling in graphs (e.g. [2][3][4][5]). A detailed survey in graph labeling is found in the paper of [1]. ...
... And we consider the two cases below: Case 1. Suppose that ≡ 1 ( 2), it is clear that there are where ∈ ( ) and ∈ ( ). Case 2. Suppose that ≡ 0( 2), it is easy to check that there are 2 of ( ) 4 ̅̅̅̅̅̅̅̅ = 2 = ( ) 4 ̅̅̅̅̅̅̅̅ , ...
... If ( ) ≡ 0( 4), then ( ) ≡ 2 ( 4). ...
Let G=(V(G), E(G)) be a connected graph with order |V(G)|=p and size |E(G)|=q. A graph G is said to be even-to-odd mean graph if there exists a bijection function phi:V(G) to {2, 4, ..., 2p} such that the induced mapping phi^*:E(G) to {3, 5, ..., 2p-1} defined by phi^*(uv)=[phi(u)+phi(v)]/2 for all uv element of E(G) is also bijective. The function is called an even-to-odd mean labeling of graph . This paper aimed to introduce a new technique in graph labeling. Hence, the concepts of even-to-odd mean labeling has been evaluated for some trees. In addition, we examined some properties of tree graphs that admits even-to-odd mean labeling and discussed some important results.
... Graphs considered here are finite, undirected and simple.Terms not defined here are used in the sense of Harary [4] and Gallian [3]. Somasundaram and Ponraj [6] introduced the concept of mean labeling of graphs. Cahit [2] introduced the concept of cordial labeling. ...
... [6] A graph G with p vertices and q edges is a mean graph if there is an injective function f from the vertices of G to 0, 1, 2, ..., q such that when each edge uv is labeled withf (u)+ f (v) 2 if f (u) + f (v) is even and f (u)+ f (v)+1 2 if f (u) + f (v)is odd then the resulting edge labels are distinct. ...
Let G be a graph and let A be a group. Let be a map. For each edge assign the label . Here o(f(u)) denotes the order of f(u) as an element of the group A. Let be the set of all integers that are labels of the edges of G. f is called a group mean cordial labeling if the following conditions hold:(1) For , where is the number of vertices labeled with x.(2) For , where denote the number of edges labeled with i.A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take A as the group of fourth roots of unity and prove that,the splitting graphs of Path and Complete Bipartite graph ( when n is even ) are group mean cordial graphs. Also we characterized the group mean cordial labeling of the splitting graph of .
... The concept of mean labeling was introduced by Somasundaram and Ponraj [12]. Different kinds of mean labeling are further studied by Gopi [9]. ...
... Particularly, interesting applications of graph labeling can be found in [1][2][3][4]. Somasundaram and Ponraj [12] have introduced the notion of mean labeling of a (p,q) graph. Different kinds of mean labeling are studied by Gayathri and Gopi in [9]. ...
The concept of mean labeling was introduced by Somasundaram and Ponraj. Different kinds of mean labeling are further studied by Gayathri and Gopi. Swaminathan and Sekar introduced the concept of modulo three graceful labeling. As an analogue Jayanthi and Maheswari introduced one modulo three mean labeling and proved that some standard graphs are one modulo three mean graphs. We have proved some necessary conditions and properties for one modulo three mean labeling and verified one modulo three meanness for some family of trees and some special graphs. In this paper, we obtain one modulo three mean labeling of some disconnected graphs.
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Babitha and Baskar proved some results on prime cordial labeling in [1]. The mean graph labeling was introduced by Somasundaram and Ponraj in [18]. Balamurugan, Thirusangu and Thomas introduced the notion of Zumkeller labeling in [2]. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
Adding new classes of integers to literature is both challenging and charming. Until a new class is completely characterized, mathematics is never going to be worth it. While it's absurd to play with integers without intended consequences. In this work, we introduce and investigate four new classes of integers namely, anti-totient numbers, half anti-totient numbers, near Zumkeller numbers and half near Zumkeller numbers by using the notion of non-coprime residues of n including n. We formulate and propose relations of these new classes of numbers with previous well-known numbers such as perfect, totient, triangular, pentagonal, and hexagonal numbers. These new classes of integers have been completely characterized. Finally, as an application of these new classes of numbers, a new graph labeling is also proposed on anti-totient numbers.
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... Also, we establish relations of anti-totient numbers with well-known classes of integers. 4,6,8,10,12,13,14,16,18,20,22, 24, 26}. The sum of the elements of S is 195 = 3·26 2 · 5. ...
... Also, also establish their relations with existing well-known classes of integers. 3,4,6,8,9,10,12,14,15,16,18,20, 21, 22}, whose sum is 180 = 3 · 24 2 · 5. If we take n = 15 then the sum of its co-prime residues is 45 which can not be written in the form of 15k for any even integer k. ...
... The graceful labelings of graphs was first introduced by Rosa in 1961 [1] and R.B. Gnanajothi introudced odd graceful graphs [2]. The concept of mean labeling was first introduced by S. Somasundaram and R. Ponraj [7]. The mean labeling of some standard graphs are studied in [5,7,8]. ...
... The concept of mean labeling was first introduced by S. Somasundaram and R. Ponraj [7]. The mean labeling of some standard graphs are studied in [5,7,8]. Further some more results on mean graphs are discussed in [6,9,10]. ...
... Later, Solomon Golomb changed the name of β-labeling to graceful. Mean labeling of graphs was rst introduced by Somasundaram and Ponraj [6]. Later, it was expanded to super mean, k-super mean, even vertex mean, odd vertex mean, even vertex odd mean, and odd vertex even mean, and several others. ...
Nowadays, the use of digital technology for communication is
constantly growing. As a result, algorithms that provide more secure communication must be employed. Despite the fact that there are numerous encryption algorithms, app developers are constantly looking for new or updated versions of the algorithms. In this proposed work, the vertex odd mean labeling method is used to produce edge values for a graph that serve as the encoded numbers for the characters of a plaintext. The complement of that graph is taken to generate a cipher graph. Communication will be safer since each text will have a distinct cipher graph. The text is encrypted and decrypted using the symmetric key technique. This algorithmic approach is useful for all texts with or without special characters.
... Chartrand et al developed the concept of radio labeling in [1]. Somasundaram S and Ponraj introduce the notion of mean labeling of graphs in [13]. Radio mean labeling was introduced by Ponraj et al in [10]. ...
Radio Even Mean Graceful Labeling of a connected graph G is a bijection from the vertex set V(G) to satisfying the condition for every . A graph which admits radio even mean graceful labeling is called radio even mean graceful graph. In this paper we investigate the radio even mean graceful labeling on degree splitting of some special graphs.
... In [17], Barrientos examined at graceful labelings of chain and corona graphs. Te mean labeling [18] of graphs was defned by Somosundaram and Ponraj, and super mean labeling [19] was discussed by Vasuki and Arockiaraj. Baskar and Arockiaraj proposed the principles of F-geometric mean labeling [20] and super geometric mean labeling [21] and highlighted them for several standard graphs. ...
Numbering a graph is a very practical and effective technique in science and engineering. Numerous graph assignment techniques, including distance-based labeling, topological indices, and spectral graph theory, can be used to investigate molecule structures. Consider the graph G , with the injection Ω from the node set to 1 , 2 , … , ∆ , where ∆ is the sum of the number of nodes and links. Assume that the induced link assignment Ω ∗ is the ceiling function of the average of root square, harmonic, geometric, and arithmetic means of the vertex labels of the end vertices of each edge. If the union of range of Ω of the node set and the range of Ω of the link set is the set 1 , 2 , … , ∆ , then Ω is called a super classical average assignment (SCAA). This is known as the SCAA criterion. In this study, the graphical structures corresponding to chemical structures based on the SCAA criterion are demonstrated. The graphical depiction of chemical substances was first defined and second, the union of any number of cycles C n , the tadpole graph, the graph extracted by identifying a line of any two cycles C m and C n , and the graph extracted by joining any two cycles by a path are all examined in this work.
... Over a period of 50 years, more than 200 labeling techniques are introduced and studied by several authors and published in various journals which are beautifully classified and indexed by Gallian in his Survey [1]. One of the miscellaneous labelings [1] called 'mean labeling' was introduced by Somasundaram et al. [11]. A graph is said to have mean labeling if there is an injective function f that maps V (G) to {0, 1, 2, . . . ...
A graph G is analytic odd mean if there exist an injective function
f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f
∗
: E → Z
such that for each edge uv with f(u) < f(v),
f
∗
(uv) =
⎧
⎨
⎩
l
f(v)
2−(f(u)+1)
2
2
m
if f (u) 6= 0;
l
f(v)
2
2
m
if f (u) = 0.
is injective. Clearly the values of f
∗ are odd. We say that f is an
analytic odd mean labeling of G. In this paper, we show that the union
and identification of some graphs admit analytic odd mean labeling by
using the operation of joining of two graphs by an edge.
... The concept of geometric mean graph was introduced by Somasundaram et al. [4] and they have investigated geometric mean labeling for some standard graphs while Somasundaram et al. [5] proved that the square of a path, crown and union of some standard graphs admit geometric mean labeling. In this paper we investigate geometric mean labeling in the context of duplication of graph elements. ...
The geometric mean labeling is a variation of arithmetic mean labeling. We investigate geometric mean labeling for various graphs resulted from the duplication of graph elements.
... A useful survey on graph labeling by J.A. Gallian (2014) can be found in [2]. Somasundaram and Ponraj [4] have introduced the notion of mean labeling of graphs. A directed graph or digraph consists of a finite set of vertices and a collection of ordered pairs of distinct vertices. ...
K. Palani et al. defined the concept of near mean labeling in digraphs. Let 𝐷 = (𝑉, 𝐴)
be a digraph where 𝑉the vertex is set and 𝐴 is the arc set. Let 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} be a 1-1 map. Define 𝑓∗: 𝐴 → {1, 2,…, 𝑞} by𝑓∗(𝑒 = ⃗𝑢⃗⃗⃗𝑣 ) = ⌈ 𝑓(𝑢)+𝑓(𝑣) /2⌉. Let𝑓∗(𝑣) =
|Σ𝑤∈𝑉 𝑓∗(⃗𝑣⃗⃗⃗𝑤⃗ ) − Σ𝑤∈𝑉 𝑓∗(⃗𝑤⃗⃗⃗⃗𝑣 )|. If𝑓∗(𝑣) ≤ 2 ∀ 𝑣 ∈ 𝐴(𝐷), then 𝑓 is said to be a near mean labeling of D and 𝐷 is said to be a near mean digraph. In this paper, different dicyclic snakes are defined and the existence of near mean labeling in them is checked.
... Somasundaram and Ponraj [9] have introduced the notion of mean labeling of a (p,q) graph. Different kinds of mean labeling are studied by Gayathri and Gopi in [6]. ...
The concept of mean labeling was introduced by Somasundaram and Ponraj [10]. Different kinds of mean labeling are further studied by Gopi [2]. Swaminathan and Sekar [11] introduced the concept of modulo three graceful labeling. As an analogue, Jayanthi and Maheswari [8] introduced one modulo three mean labeling. In [3], we obtained some necessary conditions and properties for one modulo three mean labeling. In [4-6], we obtained one modulo three mean labeling of some graphs. In this paper, we discuss one modulo three mean labeling of some path related graph.
... e origin of graph labeling called graceful labeling was characterized by Rosa in [8] and the mean labeling of graphs was introduced by Somasundram et al. in [9]. In [10], Arockiaraj et al. presented the idea of F-root square mean labeling of the graphs and examined its meanness [11]. ...
In this study, we investigate a new kind of mean labeling of graph. The ladder graph plays an important role in the area of communication networks, coding theory, and transportation engineering. Also, we found interesting new results corresponding to classical mean labeling for some ladder-related graphs and corona of ladder graphs with suitable examples.
1. Introduction and Preliminaries
All through this paper, by a graph, we mean an undirected, simple, and finite graph. For documentations and phrasing, we follow [1–6]. For a point-by-point review on graph labeling, refer [7].
Let be a path on nodes denoted by , where , and with lines denoted by , where , where is the line joining the vertices and . On each edge , erect a ladder with steps including the edge , for . The resulting graph is called the one-sided step graph, and it is denoted by . Let be a path on vertices , where and with edges , where is the line joining the vertices and . On each edge , we erect a ladder with ‘’ steps including the edge , for , and on each , we erect a ladder with steps including , for . The graph thus obtained is called the double-sided step graph, and it is denoted by . Let and be any two graphs with and vertices, respectively. Then, is the cartesian product of two graphs. A ladder graph is the graph . The graph is obtained from by attaching pendant vertices to each vertex of . The triangular ladder , for , is a graph obtained from two paths by and by adding the edges and . The slanting ladder is a graph obtained from two paths and by joining each , with . The graph having the vertices , and its edge set is .
2. Literature Survey
The origin of graph labeling called graceful labeling was characterized by Rosa in [8] and the mean labeling of graphs was introduced by Somasundram et al. in [9]. In [10], Arockiaraj et al. presented the idea of -root square mean labeling of the graphs and examined its meanness [11]. Durai Baskar and Arockiaraj talked about the -geometric meanness of some ladders in [12]. Dafik et al. researched the antimagicness of the graphs including the graph in [13]. Durai Baskar considered the logarithmic meanness in [14] and Rajesh Kannan et al. characterized idea of exponential mean graphs in [15]. In addition, more concepts of ladder graphs and related concepts have been studied in [16–24]. Recently, Muhiuddin et al. have applied various related concepts on graphs in different aspects (see, e.g., [25–31]).
3. Methodology
A labeling on a graph with vertices and edges is called a Smarandache mean labeling, for an integer and , if is injective and the induced function defined byfor all , is bijective.
Particularly, if and , such a Smarandache mean labeling is the classical mean labeling on the graph. A function is known as a classical mean labeling of a graph with nodes and edges if is injective and the incited edge assignment function characterized asfor all , is bijective. A graph that concedes a classical mean labeling is said to be classical mean graph.
A classical mean labeling of is shown in Figure 1.
... The concept of mean labeling was introduced and studied by Somasundaram and Ponraj [11]. Further some more results on mean graphs are discussed in [4,6]. ...
A graph with p vertices and q edges is said to have an even
vertex odd mean labeling if there exists an injective function
f:V(G){0, 2, 4, ... 2q-2,2q} such that the induced map
f*: E(G) {1, 3, 5, ... 2q-1} defined by f*(uv)=
f u f v
2
is a bijection. A graph that admits an even
vertex odd mean labeling is called an even vertex odd mean
graph. In this paper we pay our attention to prove some graph
operations of even vertex odd mean labeling graphs
... For notation and terminology, we follow [6] The graceful labeling of graphs was first introduced by Rosa in 1967. The concept of mean labeling was introduced and studied by Somasundaram and Ponraj [8]. Further some more results on mean graphs are discussed in [4,5]. ...
... The concept of mean labeling was introduced and studied by Somasundaram and Ponraj [11]. Further some more results on mean graphs are discussed in [4,5]. ...
... Referring to the graph labeling introduced by Gallain, a detailed survey is conducted on graph labeling [4]. Somasundaram and Ponraj [7] originated the theory of mean labeling of graphs. Many mathematicians introduced different aspects of mean labeling. ...
A function h is mentioned as a C -exponential mean labeling of a graph G V , E that has s vertices and r edges if h : V G ⟶ 1 , 2 , 3 , ⋯ , r + 1 is injective and the generated function h ∗ : E G ⟶ 2 , 3 , 4 , ⋯ , r + 1 defined by h ∗ a b = 1 / e h b h b / h a h a 1 / h b − h a , for all a b ∈ E G , is bijective. A graph which recognizes a C -exponential mean labeling is defined as C -exponential mean graph. In the following study, we have studied the exponential meanness of the path, the graph triangular tree of T n , C m P n , cartesian product of two paths P m ▫ P n , one-sided step graph of S T n , double-sided step graph of 2 S T 2 n , one-sided arrow graph of A r s , double-sided arrow graph of D A r s , and subdivision of ladder graph S L t .
... The concept of mean labeling of graphs was introduced by Somasundaram and Ponraj [10]. k-odd mean, (k, d)-odd mean labeling were introduced and discussed by Gayathri and Amuthavalli [1]. ...
The concept of mean labeling was introduced by Somasundaram and Ponraj [10]. k-odd mean, (k, d)-odd mean labeling were introduced and discussed by Gayathri and Amuthavalli [1]. k-mean, k-even mean and (k, d)-even mean labeling were further studied by Gayathri and Gopi [6]. We have introduced (k, d)-mean labeling and obtained results for some family of trees and for some special graphs [4,5]. In this paper, we investigate (k, 1)-mean labeling for some new families of graphs. Here k and d denote any positive integer greater than or equal to 1.
... Mean labeling of graphs was discussed in [10] and the concept of odd mean labeling was introduced in [9]. k-odd mean labeling and (k, d) -odd mean labeling are introduced and discussed in [5], [6], [7]. ...
... Most of the labeling methods in graph are initiated by the Rosa [9] in 1967, later -labeling was known as graceful labeling by Golomb [4] . Mean labeling was formed by Somasundaram.S and Ponraj [11] also, they showed results on gracefulness of cycle with parallel Pk chords. Gayathri.B and Gopi.R [3] , are studied about the mean graphs related Cycle. ...
In this paper, we study about the cycle with parallel P3 chords, prove that the chain of cycles 2 , (≥ 3)(≥ 2) with parallel 3 chords and Edge connected cycle 2 (≥ 3) with parallel 3 chords possess vertex even mean labeling. We prove that the two copies of even cycles 2 (≥ 3) with 3 chords joining by the path possess vertex even mean labeling. Also, we prove vertex odd mean labeling for the cycle with parallel P3 chords the chain of cycles 2 , (≥ 3)(≥ 2) with parallel 3 chords and Edge connected cycle 2 (≥ 3) with 3 chords possess vertex odd mean labeling. We show that the joining two copies of cycles 2 (≥ 3) with parallel 3 chords by the path possess vertex odd mean labeling.
... We consider only finite, undirected and simple graphs. The origin of graph labeling is graceful labeling and introduced this concept by Rosa.A [15].Afterwards many labeling was defined and few of them are harmonious labeling [7], cordial labeling [1], magic labeling [16], mean labeling [19]. Cordial analogous labeling was studied in [2,3,4,5,10,11,12,13,14,17,18]. ...
In this paper we dicuss the pair difference cordility of Mirror graph, Splitting graph, Shadow graph of some graphs. we have studied about the pair difference cordility of Mirror graph, Splitting graph, Shadow graph of some graphs.Investigation of the pair difference cordility of Mirror graph, Splitting graph, Shadow graph of some special graphs are the open problems.
... For all other standard terminology and notations we follow Harary [2]. The concept of mean labelling has been introduced by S.Somasundaram and R.Ponraj [3] in 2004. S.Somasundaram and S.S.Sandhya introduced Harmonic mean labeling [4] in 2012. ...
In this paper we contribute some new results on Power 3 mean labeling of graphs. We investigate on some standard graphs that accept Power 3 mean labeling and proved that the Line graphs of these Power 3 mean graphs are also Power 3 mean graphs. We proved that the Line graphs of Path, Cycle, Comb, ⨀ 1 , ⨀ 1,2 are Power 3 mean graphs.
... e investigation of graceful labeling is characterized by Rosa in [7] and prime labeling is defined by Tout et al. in [8]. Somasundram and Ponraj introduced the mean labeling of graphs in [9]. Durai Baskar and Arockiaraj defined the F-harmonic mean labeling [10] and discussed its meanness for some standard graphs. ...
In the present paper, we introduce the classical mean labeling of graphs and investigate their related properties. Moreover, it is obtained that the line graph operation preserves the classical meanness property for some standard graphs.
1. Introduction and Preliminaries
All through this paper, by a graph we mean a simple, undirected, and finite graph. For documentations and wording, we follow [1–5]. For a point by point review on graph labeling, we refer [6]. The line graph of a graph is defined to have as its vertices the edges of , with two being adjacent if the corresponding edges share a vertex in . The graph is obtained from by attaching pendant vertices to each vertex of . Let and be the nodes of path and copy of the star graph , respectively, then the graph is obtained from copies of and the path by joining with the central vertex of the copy of by means of an edge, for . A graph obtained by subdividing edge of by a vertex is called subdivision graph and a graph obtained from the path by replacing every edge of a path by a is called triangular snake graph .
2. Literature Survey
The investigation of graceful labeling is characterized by Rosa in [7] and prime labeling is defined by Tout et al. in [8]. Somasundram and Ponraj introduced the mean labeling of graphs in [9]. Durai Baskar and Arockiaraj defined the F-harmonic mean labeling [10] and discussed its meanness for some standard graphs. The idea of -geometric was presented by Durai Baskar et al. in [11] and -root mean labeling was presented by Arockiaraj et al. in [12] and talked about its meanness of ladder graph in [13]. Vaidya and Barasara in [14] have discussed so many results on product cordial labeling. Vaidya and Lekha in [15] presented the idea of a bi-odd sequential labeling. The labeling of L (2, 1) in [16] is researched by Prajapati and Patel. Rajesh Kannan et al. discussed the FCM labeling of graphs and its line graphs in [17]. Propelled by and crafted by such a large number of creators in the territory of graph labeling, we present another labeling called classical mean labeling. A classical mean of two positive integers need not be an integer in general. For the classical mean is to be an integer, we may use either flooring or ceiling function. In this paper, we consider only the flooring function of our discussion and try to analyze that the line graph operation preserves the classical meanness property for some standard graphs. The labeling is one of the well studied area in Graph Theory. So, we are interested in defining new labeling called classical mean labeling. A classical mean labeling is for getting more accuracy of all the edge labeling by using the average of four different types of means of the vertex labeling of the given graph. Recently, Muhiuddin et al. studied various related concepts on graphs (see [18–22]).
Graph labeling assumes an essential job in different areas of the real world system. The concepts of classical mean labeling are utilized to demonstrate numerous kinds of processes and relations in biological, social, material physical, and data systems. It is a powerful tool that makes complicated patterns to be learned easily and conveniently in various fields. A static network can be represented as a specific kind of graph by connecting nodes in some topology, and labeling can be applied for automatic routing of data in a network. The graph can be cycle, path, circuit, walk, and connected which represent a fixed network. For each network, labeling is done with a constant which helps routing to automatically detect next node in the network. The classical mean labeling is used in fast communication in sensor networks for finding the more accuracy level of sensor units.
3. Methodology
A function is known as a classical mean labeling of a graph with nodes and edges if is injective and the incited edge assignment function characterized asfor all , is bijective. From Figure 1, a graph that concedes a classical mean labeling is said to be classical mean graph.
... Mean labeling was introduced by S. Somasundaram and R. Ponraj [6][7]. Relaxed mean labeling was introduced by V. Maheswari, D.S.T. Ramesh and V. Balaji [4][5]. ...
... Several types of graph labeling and a detailed survey is available in [2]. S. Somasundaram and R. Ponraj [3] have introduced the notion of mean labeling of graphs. A graph G with p vertices and q edges is called mean graphif there is an injective function f from the vertices of G to {0, 1, …, q} such that when each edge uv is labeled with if f(u)+f(v) is even and with if f(u)+f(v) is odd, then the resulting edge labels are distinct. ...
In this paper, we introduce a new labeling called mean square sum labeling. A bijection f: →{0,1, ..., p-1} G is said to be a mean square sum labeling if the induced function f*:E(G)→N given by f*(uv) = or for every uv is injective. A graph which admits a mean square sum labeling is called a mean square sumgraph. In this paper we prove that Path, Comb, Star graph, Cycle, Bistar, Doublestar, G = K 2 +mK 1 , Ladder, P n ʘK 2 and some more graphs are mean square sum graphs.
Fruits play an essential role in our diets, but diseases affecting fruit health can lead to
significant global economic losses. Among fruit crops, apples are particularly important, ranking second in
global fruit production. In 2017, global apple production reached 83.1 million tons, underlining their
widespreadconsumption and importance. Apples are enjoyed fresh, but their versatility also allows them to
be used in a wide range of products, including juices, ciders, applesauce, and dried apples. Approximately
33% of the apples produced worldwide are processed into these various products. Despite their global
significance, apple fruit diseases pose a major threat to yield and quality. To address this issue, the
development of an automatedsystem to detect and classify apple diseases is essential. Recent advancements in
deep learning have providedpowerful tools for tackling complex image recognition tasks, making them ideal
for identifying apple diseases with accuracy. This research introduces the VGG-SENet model, a method
designed to identify and classify apple diseases. The core of the VGG-SENet model combines a tuned VGG16
architecture with an ensemble of Squeeze-and-Excitation (SE) blocks. The tuned VGG16 provides a robust
feature extraction capability, while the ensemble of SE-blocks enhances the model's ability to focus on the
most important features for disease detection through a channel-wise attention mechanism. This combination
allows the model to effectively distinguish between healthy apples and those affected by common diseases
like blotch, rot, and scab. The VGG-SENet model leverages these techniques to capture and emphasize the
discriminative featuresthat are most indicative of disease, leading to superior performance in classification
tasks. Rigorous testing on apple disease datasets has shown that VGG-SENet outperforms other deep learning
models, achieving high accuracy and strong performance across various evaluation metrics. Overall, the
VGG-SENet model holds significant potential as a decision-support tool for farmers, enabling them to
quickly and accurately identify apple diseases. This early detection capability can improve crop management
and help reduce the economic impact of apple diseases on global production.
Keywords: Apple fruit diseases, VGG-SENet, Deep learning, Image classification
This work introduces the principle of “an even point (vertex) odd ratio (mean) labeling, which is specifically applied to a graph ‘G’ consisting of ‘p’ vertices and ‘q’ edges. Even point (vertex) odd ratio (mean) labeling is exhibited by a graph G in the presence of an injectionbased function f : V of G → {0, 2, 4, ... 2q – 2, 2q} ensuring that the function derived from it (induced map) g* : E of G→{1, 3, 5, ... 2q – 1} specified by g* (uv) = g(u)+g(v)/2 is a bijection. Graphs that meet these criteria are termed an even point (vertex) odd ratio (mean) graphs. This paper explores the properties of an even point (vertex) odd ratio (mean) labeling in various graph structures.
Edited Book titled Trends in Contemporary Mathematics & Applications provides an excellent international platform for academicians, and researchers around the world to publish their research and to exchange ideas on recent developments in Contemporary Mathematics & Its Applications. Its goal is to support researchers through the dissemination of information, research findings, and practice. This book facilitates the publication of original articles, reports of professional experience, comments, and reviews.
The concept of mean labeling was introduced by S. Somasundaram and Ponraj in 2003. Many research papers have published in this topic. In this paper we have established a general format for labeling of .
This research work focuses on the strategies for coding a text message is enunciated by changing plain text into cipher text through graph lebeling and GMJ coding. Illustrations are provided using three star graphs K1,λ ∧ K1,μ ∧ K1,θ by applying the mean labeling to it at two altered situations θ = λ + μ when λ > 9 and θ = λ + μ − 7 when λ > 9.
Let h : V (G) → S 3 be a function defined in such a way that for every edge uv ∈ E(G), |o(h(u)) − o(h(v))| = 1. The function h is called a group S 3 cordial difference labeling if |v h (i)−v h (j)| ≤ 1 for every i, j ∈ S 3 , i = j, where v h (j) denote the number of vertices of G having label j under h. A graph G which admits a group S 3 cordial difference labeling is called a group S 3 cordial difference graph. In this paper, we prove that double alternate quadrilateral snake, crown, L n K 1 , subdivision of comb, subdivision of bistar and subdivision of star are a group S 3 cordial difference graphs.
In this article, we come across for non-trivial integer solutions to the negative Pell equation 𝛼^2=113𝛽^2−13^𝜆, 𝜆∈𝑁 for the singular choices of 𝜆 scrupulous by (𝑖) 𝜆=1,(𝑖𝑖) 𝜆=3,(𝑖𝑖𝑖) 𝜆=5,(𝑖𝑣) 𝜆=2𝜇,(𝑣) 𝜆=2𝜇+5,∀ 𝜇∈𝑁.
Let G be a (p, q) graph and f : V(G) → {0, 2, 4, … , 2q - 2, 2q} be an injection. For each edge e = uv the induced edge labeling f* : E(G) → {1, 3, 5, … , 2q - 1} defined by is a bijection. Then f is called even vertex odd mean labeling if f (V(g)) ∪ { f*(e) : e ∈ E(G)} = {0, 1, 2, 3, … , 2q}. A graph that admits an even vertex odd mean labeling is called even vertex odd mean graph. Here we prove that (Cm; Cn), [Cm; Cn] and are even vertex odd mean graphs for all m, n ≡ 0 (mod 4) and [2Pm; Cn], Sm(Cn) are even vertex odd mean graphs for all m ≥ 1, n ≡ 0 (mod 4).
In this paper, we investigate Smarandache curves according to type-2 Bishopframe in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3-space are presented. Besides, we illustrate examples of our results.
Let G=(V,E) be a graph with p vertices and q edges. A graph G is said to have an even vertex odd mean labeling if there exists a function f:V(G)→{0,2,4,…,2q} satisfying f is 1-1 and the induced map f^*:E(G)→{1,3,5,…,2q-1} defined by f^* (uv)=(f(u)+f(v))/2 is a bijection. A graph that admits even vertex odd mean labeling is called an even vertex odd mean graph. In this paper, we prove that Tp-tree T , T@P_n , T@〖2P〗_n and 〈TõK_(1,n) 〉 ,where T is a Tp-tree, are even vertex odd mean graphs.
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