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Super Root Square Mean Labeling of Some Graphs
... Sandhya proved that double triangular snake, alternate double triangular snake, double quadrilateral snake and alternate double quadrilateral snake graphs are super root square mean graphs [8]. Devi and Kumar proved that m copies of path , some copies of complete graph, corona product of path and complement of complete graph 2 , middle graph of path, and dragon graph are super root square mean graphs [9]. ...
Let G be a graph with q edges. A labelling f of G is said to be root square mean labelling if f : V ( G )U E ( G )→{1,2,…, q +1} such that when each edge e = uv labelled with f ( e )=[ f ( u ) ² + f ( v ) ² /2] or f ( e )=[ f ( u ) ² + f ( v ) ² /2] then the resulting edge labels are distinct. A graph G is called a root square mean graph if G can be labelled by a root square mean labelling. In this paper we determine a root mean square labelling of two graphs obtained from path, which are corona product of ladder and complete graph with order 1, and a graph obtained from triangular snake by join one vertex with degree 2 in each triangle to a new vertex. The method of labelling construction is we need to do labeling to the vertices of the graph with label 1, 2, 3, …, q+ 1. The labels of the vertices are not necessarily different. The next step is we need to do labeling to the edges with the certain formula by using the vertex labeling. The edge labels must be different. By the labelling we construct, we proof that the two graphs are root square mean graphs.
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.
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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- F Harary
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Root square mean labeling of graphs
- S S Sandhya
- S Somasundaram
- S Anusa
S.S. Sandhya, S. Somasundaram and S. Anusa, Root square
mean labeling of graphs, Int. J. Contemp. Math. Sciences, Vol.9,
14(2014), 669-676.
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