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A graph G=(V,E) with p vertices and q edges is called a harmonic mean graph if it is possible to label the vertices x∈V with distinct labels f(x) from 1,2,...,q+1 in such a way that when each edge e=uv is labeled with f(uv)=2f(u)f(v) f(u)+f(v) or 2f(u)f(v) f(u)+f(v) then the edge labels are distinct. In this case f is called harmonic mean labeling...

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The concept of vertex equitable labeling was introduced in [9]. A graph $G$ is said to be vertex equitable if there exists a vertex labeling $f$ such that for all $a$ and $b$ in $A$, $\left|v_f(a)-v_f(b)\right|\leq1$ and the induced edge labels are $1, 2, 3,\cdots, q$. A graph $G$ is said to be a vertex equitable if it admits a vertex equitable lab...

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... The labeling is referred to as a vertex labeling (or edge labeling) if the set of vertices (or edges) represents the mapping's domain. The idea of Mean labelling of graphs was first presented by Somasundaram and Ponraj [4][5]. Labelling with a Harmonic Mean was first proposed by S.S. Sandhya and S.D. Deepa [6]. ...
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We describe a function as – Power 3. If constitute both the induced edge labelling and take be an injective function and express it as, then a graph's Heronian Mean Labelling with p nodes and q lines is or with distinct edge labels. In this manuscript we have proved the – Power -3 Mean labeling behaviour of Path, Twig Graph, Triangular ladder . We have also investigated - Super power -3 Heronian Mean labelling of graghs. Also, we prove that is not – Power - 3 Heronian Mean graph and - Super power -3 Heronian Mean labelling of Snake related graphs like triangular, alternative triangular and double triangular snake graphs.
... Some of the harmonic mean graphs are investigated by S. Meena and M. Sivasakthi in [4]. The concept of Harmonic Mean labeling of graph was introduced by S. Somasundaram, R. Ponraj and S.S. Sandhya [5,6] and they investigated the existence of Harmonic mean labeling of several family of graphs such as this concept was then studied by several authors and studied their behavior in [7], [8], [9], and [10]. ...
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A graph G with p vertices and q edges is called a harmonic mean(HM) labeling if it is possible to label the vertices x∈v with distinct labels ρ(x) from {1,2,⋯,q+1} in such a way that each edge e=ab is labeled with ρ(ab)=⌈(2ρ(a)ρ(b))/(ρ(a)+ρ(b))⌉ or ⌊(2ρ(a)ρ(b))/(ρ(a)+ρ(b))⌋ then the edge labels are distinct.In this case ρ is called Harmonic mean(HM) labeling of G. In this paper we introduce new graphs obtained from triangular snake graph TS_n such as TS_n∘K_1, and prove that they are Harmonic Mean labeling graphs.
... Gnanajothi R. B. introduced the odd graceful graph in [2]. Mean labeling was established, and Somasundaram S and Ponraj R explored the meanness of various conventional graphs [5][6][7] [8]. Odd mean labeling of a graph was initially introduced by K. Manickam and M. Marudai [9]. ...
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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.
... The terms not defined here are used in the sense of Harary [1]. To fetch up the PDML we did survey on labeling of graphs by using [2].Further reading on [4], [5], [6], which gives the result by defining PDML. III. ...
... A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. S.Somansundaram and R. Ponraj introduced mean labeling of graphs in [3] and further studied in [4]. ...
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The concept of one modulo three mean labeling graph is, if there is an injective function φ from the vertex set of G to the set { a /0 ≤ a ≤ 3 q − 2 and either a ≡ 0( mod 3) or a ≡ 1 ( mod 3)} where q is the number of edges of G and φ induces a bijection φ * from the edge set of G to { a /1≤ a ≤ 3 q − 2, a ≡ 1 ( mod 3) } given by φ * ( u υ ) = [ φ ( u ) + φ ( υ ) 2 ] and the function φ is called one modulo three mean labeling of G. In this paper, we obtain the results of one modulo three mean labeling of some several graphs.
... 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]. ...
... They investigated the existence of harmonic mean labeling of several family of graphs studied by several authors. We have proved Harmonic mean labeling of subdivision graphs such as  1 ,  2 , H-graph, crown, [6,7].The following definitions are useful for the present investigation. ...
... S.Somasundaram, R.Ponraj and S.S.Sandhya were introduced the concept of harmonic mean labeling of graphs. They investigated the existence of harmonic mean labeling of several family of graphs such as path, comb, cycle C n , complete graph K n complete bipartite graph K 2,2 , triangular snake T n , quadrilateral snake Q n , alternate triangular snake A(T n ), alternate quadrilateral snake A(Q n ), crown C n K 1 ,C n K 2 ,C n K 3 , dragon, wheel in [8][9][10]. The harmonic mean labeling of step ladder S(T n , P n K 2 ,C n K 2 , flower graph, L n K 2 , triangular ladder, double triangular snake D(T n ), alternate double triangular snake A(DT n ) double quadrilateral snake D(Q n ), alternate double quadrilateral snake A(DQ n ), Q n K 1 , (C m K 1 ) ∪ C n , (C m K 1 ) ∪ P n , are investigated by C.Jayasekaran, C.David raj and S.S.Sandhya [11,12]. ...
... Definition 1.1.[8] A Graph G = (V, E) with p vertices and q edges is called a Harmonic mean graph if it is possible to label the vertices v ∈ V with distinct labels f (v) from {1, 2, ..., q + 1} in such a way that when each edge e = uv is labeled with ...
... 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]. ...
... Suppose if ( ) = 8 fu , then if the adjacent vertex takes the minimum possible label, that is 0, the incident edge will get the edge label 4. If the adjacent vertex is assigned the maximum possible label, 6, then that incident edge will get the label 7. Therefore the overall possible edge labels that 8 can generate are 4,5,6,7 . That is ( ) = 8 fu generates only four edge labels which is not sufficient to label 6 edges distinctly. ...
... Suppose if ( ) = 10 fu , then if the adjacent vertex takes the minimum possible label, that is 0, the incident edge will get the edge label 5. If the adjacent vertex is assigned the maximum possible label 8, then that incident edge will get the label 9. Therefore the overall possible edge labels that 10 can generate are 5,6,7,8,9 . That is ( ) = 10 fu generates only 5 edge labels which is not sufficient to label 7 edges distinctly. ...
... Suppose if ( ) = 8 fu , then if the adjacent vertex takes the minimum possible label, that is 0, the incident edge will get the edge label 4. If the adjacent vertex is assigned the maximum possible label, 10, then that incident edge will get the label 9. Therefore the overall possible edge labels that 8 can generate are 4,5,6,7,8,9 . That is ( ) = 8 fu generates only 6 edge labels which is not sufficient to label 7 edges distinctly. ...