The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367
LINE GRAPH OF ROOT SQUARE MEAN GRAPHS Sandhya S.S1 , Aswathy.H2 Author 1 : Research Supervisor, Department of Mathematics, SreeAyyappa College for Women, Chunkankadai. Author 2 : Research Scholar SreeAyyappa College for Women, Chunkankadai. [Affiliated to ManonmaniamSundararanar University, Abishekapatti – Tirunelveli - 627012, Tamilnadu, India] Email: [email protected] [email protected]. Abstract A graph G = (V,E) with p vertices and q edges is said to be a Root Square 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
푓(푢)2+푓(푣)2 푓(푢)2+푓(푣)2 f (e=uv) = f(uv) = ⌈√ ⌉ or ⌊√ ⌋, then the resulting edge labels are distinct. In 2 2 this case f is called a Root Square Mean labeling of G . In this paper we prove Root Square Mean labeling of some Line graphs.
Key words: Graphs, Labeling, Mean Labeling, Root Square Mean Labeling, Line graph. Introduction The graph considered here will be finite, undirected and simple. The vertex set is denoted by V(G) and the edge set is denoted by E(G). For all detailed survey of graph labelling, we refer to Galian [1]. For all other standard terminology and notations we follow Harary[2]. S.S.Sandhya, S.Somasundaram and S. Anusa introduced the concept of Root Square Mean labeling of graphs in [3]. In this paper we prove Root square mean labeling of some Line graph. Definition 1.1 : A graph G = (V,E) with p vertices and q edges is said to be a Root Square 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
푓(푢)2+푓(푣)2 푓(푢)2+푓(푣)2 f (e=uv) = f(uv) = ⌈√ ⌉ or ⌊√ ⌋, then the edge labels are distinct. In this case, f 2 2 is called a Root Square Mean labeling of G. Definition 1.2 : Let G = (V,X) be a graph with X ≠ 휙. Then X can be thought of as a family of 2 element subset of V. The intersection graph Ω(X) is called the Line Graph of G and is denoted by
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L(G).Thus the points of L(G) are the lines of G and two points in L(G) are adjacent iff the corresponding lines are adjacent in G. Example 1.3:
G L(G) Diamond Graph Definition 1.4 : The Diamond graph is a planar undirected graph with 4 vertices and 5 edges.
G L(G) Theorem 1.5 :
Line graph of Diamond graph L(Gd) is a Root Square Mean Graph. Proof:
Let G = L(Gd). Let the vertex set of G be {ui ; 1 ≤ i ≤ 5} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 4}∪ {u1ui ; 3 ≤ i ≤ 5}∪ {u2u5}. Define a function : f:V(G) 1,2,………..q+1 by
f(ui) = 1
f(ui+1) = 2i ; 1 ≤ i ≤ 3
f(u5) = 9
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Then the edges are labeled with
f(uiui+1) = 2i-1 ; 1 ≤ i ≤ 3
f(u4u5) = 8
f(u1ui+2) = 2i ; 1 ≤ i ≤ 3
f(u2u5) = 7 By the above labeling pattern , f is a Root square mean labeling.
Example 1.6 : The labeling pattern of L(Gd) is shown below
L(Gd)
Bull graph Definition 1.7 : The Bull graph is a planar undirected graph with 5 vertices and 5 edges in the form of a triangle with two disjoint pendent edge.
G L(G) Theorem 1.8 :
Line graph of Bull graph L(Gb)is a Root Square Mean Graph. Proof:
Let G = L(Gb). Let the vertex set of G be {ui ; 1 ≤ i ≤ 5} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 4}∪ {u1ui ; 3 ≤ i ≤ 5}.
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Define a function: f :V(G) 1,2,……….q+1 by
f(ui) = i ; 1 ≤ i ≤ 2
f(ui+1) = 2i ; 2 ≤ i ≤ 4 Then the edges are labeled with
f(uiui+1) = 2i-1 ; 1 ≤ i ≤ 4
f(u1ui+2) = 2i ; 1 ≤ i ≤ 3 By the above labeling pattern , f is a Root square mean labeling .
Example 1.9 : The labeling pattern of L(Gb) is shown below
L(Gb) Fork graph : Definition 1.10 : The Fork Graph sometimes also called the chair graph, is the 5 vertices tree and it has 4 edges
G L(G) Theorem 1.11 :-
Line graph of Fork graph L(Gf) is a Root Square Mean Graph.
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Proof:
Let G = L(Gf). Let the vertex set of G be {ui ; 1 ≤ i ≤ 4} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 3}∪ {u1u3}. Define a function: f :V(G) 1,2,……….q+1 by
f(ui) = i; 1 ≤ i ≤ 4 Then the edges are labeled with
f(uiui+1) = i ; 1 ≤ i ≤ 2
f(u3u4) = 4 , f (u1u3) = 3 By the above labeling pattern , f is a Root square mean labeling.
Example 1.12 : The labeling pattern of L(Gf) is shown below
L(Gf) Fish graph: Definition 1.13: Fish graph is the graph on 6 vertices and 7 edges.
G L(G)
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Theorem 1.14 :
Line graph of Fish graph L(Gfh)is a Root Square Mean Graph. Proof:
Let G = L(Gfh). Let the vertex set of G be {ui ; 1 ≤ i ≤ 7} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 6}∪ {u3u6,u3u7,u4u6,u4u7,u1u7}. Define a function f: v (G) 1,2, ……….. q +1
f(ui) = 2i-1 ; 1 ≤ i ≤ 6
f(u7) = 2 Then the edges are labeled with
f(uiui+1) = 2i ; 1 ≤ i ≤ 3
f(uiui+1) = 2i+1 ; 4 ≤ i ≤ 5
f (u6u7) = 7 , f (u3u6) = 8 , f (u3u7) = 3
f (u4u6) = 10 , f (u4u7) = 5 , f (u1u7) = 1 By the above labeling pattern , f is a Root square mean labeling.
Example 1.15: The labeling pattern of L(Gfh) is shown below
L(Gfh)
Cross Graph Definition 1.16: The cross graph is the 6 vertex tree and it has 5 edges.
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G L(G) Theorem 1.17 :
Line graph of Cross graph L(Gc)is a Root Square Mean Graph.
Proof :
Let G = L(Gc). Let the vertex set of G be {ui ; 1 ≤ i ≤ 5} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 4}∪ {u2u4,u3u5}. Define a function : f : V(G) 1,2, ……….. q +1
f(ui) = i ; 1≤ i ≤ 4
f (u5) = 6 Then the edges are labeled with
f(uiui+1) = i ; 1 ≤ i ≤ 3
f(u4u5) = 6
f(u2u4) = 4
f(u3u5) = 5 By the above labeling pattern , f is a Root square mean labeling.
Example 1.18: The labeling pattern of L(Gc) is shown below
L(Gc)
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Cricket Graph Definition 1.19: The cricket graph is the 5 vertex graph.
G L(G)
Theorem 1.20 :
Line graph of Cricket graph L(Gct)is a Root Square Mean Graph. Proof :
Let G = L(Gct). Let the vertex set of G be {ui ; 1 ≤ i ≤ 5} and the edge set of G be {uiui+1 ; 1 ≤ i ≤ 4}∪ {u1ui+2 ; 1 ≤ i ≤ 2}∪ {u2u4,u3u5}. Define a function : f : V(G) 1,2, ……….. q +1
f(u1) = 1
f(ui+1) = 2i ; 1≤ i ≤ 4 From the above labeling pattern the edge labels are all distinct .
Hence L(Gct) is a Root square mean graph.
Example 1.21 : The labeling pattern of L(Gct) is shown below
L(Gct)
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Conclusion: All Line graphs are not Root Square mean graphs. It is very interesting to investigate graphs which admit Root Square mean labeling. In this paper we proved that Line graph of Diamond graph, Bull graph, Fork graph, Fish graph, Cross graph and Cricket graph are Root Square mean graphs. It is possible to investigate similar results for several other graphs. References [1] Galian,J.A.(2012) A Dynamic Survey of Graph Labeling.The Electronic Journal of combinatories. [2] Harary,F.(1988)Graph Theory.Narosa Publishing House Reading,New Delhi. [3] S.S.Sandhya, S.Somasundaram, S.Anusa, Root Square Mean Labeling of Graphs International Journal of Contemporary Mathematical Sciences, Vol.9,2014,no.14, 667-676. [4] M.Venkatachalapathy, K.Kokila and B.Abarna.(2016)Some Trends In Line Graphs. Advances in Theoretical and Applied Mathematics,Volume 11,no.2,171-178.
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