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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.

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... Te circumcoronene series of benzenoids (Hm) constituted the subject of a study by Siddique et al. that established the topological indices of the double and strong double graphs in [14]. Vaidya et al. highlighted the harmonic and geometric mean labeling for diferent graphs that was generated by the duplication of graph elements in [11,15]. Durai Baskar et al. thoroughly summarised the F-geometric meanness of graphs [16,17], which can be obtained by duplicating any edge of the graph. ...
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A graph admits flooring function of centroidal meanness property if its injective node assignment Λ is from 1 to 1+q along with bijective link assignment Λ⋆uv=⌊1/3Γu,vΛu2+Γu,v2+Λv2⌋, where Γu,v=Λu+Λv, is from 1 to q. We examined the flooring function of centroidal meanness values of a few duplicates yielded graphs in this context.
... A ladder L n , n ≥ 2, is the graph P 2 ×P n . Duplicating of an edge e = uv of a graph G produces a new graph G ′ by adding an edge e ′ = u ′ v ′ such that N (u ′ ) = (N (u)∪{v ′ })−{v} and N (v ′ ) = (N (u)∪{u ′ })−{u} [6]. ...
... Arockiaraj et al. introduced the concept of F-root square mean graphs [11] and evaluated the meanness of some graphs by duplicating graph components [12]. Vaidya and Barasara thoroughly examined the harmonic mean [8] and geometric mean [13] labeling for a variety of graphs emerging from graph element duplication. In [14], Maya and Nicholas researched the duplication of divisor cordial graphs. ...
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Classical mean labeling of a graph G with p vertices and q edges is an injective function from the vertex set of G to the set 1, 2, 3,. .. , q + 1 such that the edge labels obtained from the ooring function of the average of mean of arithmetic, geometric, harmonic, and root square of the vertex labels of each edge's end vertices is distinct, and the set of edge labels is 1, 2, 3,. .. , q. One of the graph operations is to duplicate the graph. e classical meanness of graphs formed by duplicating an edge and a vertex of numerous standard graphs is discussed in this study.
... A ladder L n , n ¥ 2, is the graph P 2 lP n , the Cartesian product of the graphs P 2 and P n . The graph G ¥ K 1 is obtained from the graph G by attaching a new pendant vertex at each vertex of G. Duplicating of an edge e uv of a graph G produces a new graph G I by adding an edge e I u I v I such that N pu I q N puq tv I u ¡ tvu and N pv I q N pvq tu I u ¡ tuu [9]. ...
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An injective function f:V(G){0,1,2,,q}f:V(G)\rightarrow \{0,1,2,\dots,q\} is an odd sum labeling if the induced edge labeling ff^* defined by f(uv)=f(u)+f(v),f^*(uv)=f(u)+f(v), for all uvE(G),uv\in E(G), is bijective and f(E(G))={1,3,5,,2q1}.f^*(E(G))=\{1,3,5,\dots,2q-1\}. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.
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Graph labeling
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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 of G. The concept of Harmonic mean labeling was introduced in [the authors, “Harmonic mean labelings of graphs” (to appear)]. In [loc. cit.] and in [Int. J. Contemp. Math. Sci. 7, No. 1–4, 197–208 (2012; Zbl 1251.05128)] we investigate the harmonic mean labeling of several standard graphs such as path, cycle comb, ladder, triangular snakes, quadrilateral snakes etc. In the present paper, we investigate the harmonic mean labeling for a polygonal chain, square of the path and dragon. Also we enumerate all harmonic mean graph of order ≤5.
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A vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy a label depending on the vertex labels f(x) and f(y). The two best known labeling methods are called graceful and harmonious labelings. A function f is called a graceful labeling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, . . . , q} such that, when each edge xy is assigned the label |f(x) - f(y)|, the resulting edge labels are distinct. A function f is called a harmonious labeling of a graph G with q edges if it is an injection from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y) (mod q), the resulting edge labels are distinct. When G is a tree, exactly one label may be used on two vertices. Over the past three decades many variations of graceful and harmonious labelings have evolved and about 300 papers have been written on the subject of graph labeling....
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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.
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A graphs G=(V,E) with p vertices and q edges is said to be a geometric mean graph if it is possible to label the vertices x∈V with distinct labels f(x) from 1, 2,⋯,q+1 in such way that when each edge e=uv is labeled with f(uv)=f(u)f(v) or f(u)f(v) then the edge labels are distinct. Here we prove that C m ∪P n , C m ∪C n , nK 3 ,nK 3 ∪P n , nK 3 ∪C m , crown, square of a path are geometric mean graphs.