Asia Pacific Journal of Research
Vol: I. Issue XXXV, January 2016
ISSN: 2320-5504, E-ISSN-2347-4793
SUPER ROOT SQUARE MEAN LABELING OF SOME NEW GRAPHS
1
S.S.Sandhya
2
S.Somasundaram and
3
S.Anusa
1.Department of Mathematics,Sree Ayyappa College for women,Chunkankadai:629003,
2.Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli:627012, 3.Department of
Mathematics, Arunachala College of Engineering for Women,Vellichanthai-629203,
ABSTRACT
Let 𝐺 be a 𝑝, 𝑞 graph and 𝑓: 𝑉(𝐺) → 1,2,3, … , 𝑝 + 𝑞 be an injective function. For each edge = 𝑢𝑣 , let
𝑓 ∗ 𝑒 = 𝑢𝑣 =
𝑓(𝑢)2 +𝑓(𝑣)2
2
or
𝑓(𝑢)2 +𝑓(𝑣)2
2
, then 𝑓 is called a
super root square mean labeling if 𝑓 𝑉 ∪
𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺 = 1,2, … , 𝑝 + 𝑞 . A graph that admits a super root square mean labeling is called as super root
square mean graph. In this paper we prove that 𝑃𝑛 ⨀𝐾1,2 , 𝑃𝑛 ⨀𝐾1,3 , 𝑇𝐿𝑛 , 𝑇𝑛 ⨀𝐾1 , 𝑄𝑛 ⨀𝐾1 are super root square mean
graphs.
Key Words:
Root Square mean graph, Super Root Square mean graph, Path, Triangular snake, Quadrilateral snake.
1. Introduction
All graphs in this paper are finite, simple and undirected graph 𝐺 = 𝑉, 𝐸 with 𝑝 vertices and 𝑞 edges. For all
detailed survey of graph labeling we refer to Gallian [1]. For all other standard terminology and notations we follow
Harary [2]. The concept of Root Square mean labeling was introduced by S.S.Sandhya, S.Somasundaram and S.Anusa in
[3] and proved many results in [4,5,6,7,8,9]. In this paper we proved that 𝑃𝑛 ⨀𝐾1,2 , 𝑃𝑛 ⨀𝐾1,3 , 𝑇𝐿𝑛 , 𝑇𝑛 ⨀𝐾1 , 𝑄𝑛 ⨀𝐾1 are
super root square mean graphs. The following definitions and theorems are useful for the present study.
Definition 1.1: Let 𝐺 be a 𝑝, 𝑞 graph and 𝑓: 𝑉(𝐺) → 1,2,3, … , 𝑝 + 𝑞 be an injective function. For each edge = 𝑢𝑣 , let
𝑓 ∗ 𝑒 = 𝑢𝑣 =
𝑓∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
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𝑓(𝑢)2 +𝑓(𝑣)2
2
or
𝑓(𝑢)2 +𝑓(𝑣)2
2
, then 𝑓 is called a
super root square mean labeling if 𝑓 𝑉 ∪
= 1,2, … , 𝑝 + 𝑞 .
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Asia Pacific Journal of Research
Vol: I. Issue XXXV, January 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Definition1.2: A Triangular Ladder is a graph obtained from 𝐿𝑛 by adding the edges 𝑢𝑖 𝑣𝑖+1 ,
1 ≤ 𝑖 ≤ 𝑛 − 1 , where 𝑢𝑖 and 𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑛 are the vertices of 𝐿𝑛 such that 𝑢1 𝑢2 ⋯ 𝑢𝑛 and 𝑣1 𝑣2 ⋯ 𝑣𝑛 are two paths of
length 𝑛 in the graph 𝐿𝑛 .
Definition1.3: A Triangular snake 𝑇𝑛 is obtained from a path 𝑢1 𝑢2 ⋯ 𝑢𝑛 by joining 𝑢𝑖 and 𝑢𝑖+1 to a new vertex 𝑣𝑖 for
1 ≤ 𝑖 ≤ 𝑛 − 1.
Definition1.4: A Quadrilateral snake 𝑄𝑛 is obtained from a path 𝑢1 𝑢2 ⋯ 𝑢𝑛 by joining 𝑢𝑖 and 𝑢𝑖+1 to two new vertices 𝑣𝑖
and 𝑤𝑖 1 ≤ 𝑖 ≤ 𝑛 − 1 respectively and then joining 𝑣𝑖 and 𝑤𝑖 .
Definition1.5: The Corona of two graphs 𝐺1 and 𝐺2 is the graph 𝐺 = 𝐺1 ⨀𝐺2 formed by taking one copy of 𝐺1 and
𝑉(𝐺1 ) copies of 𝐺2 where the ith vertex of 𝐺1 is adjacent to every vertex in the ith copy of 𝐺2 .
Theorem 1.6: Any path 𝑃𝑛 is a Super Root Square mean graph.
Theorem 1.7: Ladder 𝐿𝑛 is a Super Root Square mean graph.
Theorem 1.8: Triangular Snake 𝑇𝑛 is a Super Root Square mean graph.
Theorem 1.9: Quadrilateral Snake 𝑄𝑛 is a Super Root Square mean graph.
2.Main Results
Theorem 2.1: Triangular Ladder 𝑇𝐿𝑛 is a Super Root Square Mean graph.
Proof: Let 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 and 𝑣1 , 𝑣2 , ⋯ , 𝑣𝑛 be two paths of length 𝑛. Join 𝑢𝑖 and 𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑛. Join 𝑢𝑖 and 𝑣𝑖+1 , 1 ≤
𝑖 ≤ 𝑛 − 1.The resulting graph is 𝑇𝐿𝑛 .
Define a function 𝑓: 𝑉(𝑇𝐿𝑛 ) → 1,2, … , 𝑝 + 𝑞 by
𝑓 𝑢1 = 1 , 𝑓 𝑢𝑖 = 6𝑖 − 6 , 2 ≤ 𝑖 ≤ n
𝑓 𝑣𝑖 = 6𝑖 − 3 , 1 ≤ 𝑖 ≤ n
Then the edges are labeled as
𝑓 𝑢𝑖 𝑢𝑖+1 = 6𝑖 − 2 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑣𝑖 𝑣𝑖+1 = 6𝑖 + 1 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑣𝑖 = 6𝑖 − 4 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑣𝑖+1 = 6𝑖 − 1 , 1 ≤ 𝑖 ≤ 𝑛 − 1
Then 𝑓 𝑉 ∪ 𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
= 1,2, … , 𝑝 + 𝑞 . Hence by definition 1.1 ,Triangular Ladder 𝑇𝐿𝑛 is a Super root
square mean graph.
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Asia Pacific Journal of Research
Vol: I. Issue XXXV, January 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Example 2.2: The labeling pattern of 𝑇𝐿5 is shown below.
Figure 1
Theorem 2.3: 𝑃𝑛 ⨀𝐾1,2 is a super root square mean graph.
Proof: Let 𝑢1 , 𝑢2 , … , 𝑢𝑛 be the path 𝑃𝑛 . Let 𝑣𝑖 and 𝑤𝑖 , 1 ≤ 𝑖 ≤ 𝑛 be the vertices of 𝐾1,2 attached to 𝑢𝑖 .
Define a function 𝑓: 𝑉(𝑃𝑛 ⨀𝐾1,2 ) → 1,2, … , 𝑝 + 𝑞 by
𝑓 𝑢𝑖 = 6𝑖 − 3 , 1 ≤ 𝑖 ≤ n
𝑓 𝑣𝑖 = 6𝑖 − 5 , 1 ≤ 𝑖 ≤ n
𝑓 𝑤𝑖 = 6𝑖 − 1 , 1 ≤ 𝑖 ≤ n
Then the edges are labeled as
𝑓 𝑢𝑖 𝑢𝑖+1 = 6𝑖, 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑣𝑖 = 6𝑖 − 4 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑢𝑖 𝑤𝑖 = 6𝑖 − 2 , 1 ≤ 𝑖 ≤ 𝑛
Then 𝑓 𝑉 ∪ 𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
= 1,2, … , 𝑝 + 𝑞 . Hence by definition 1.1, 𝑃𝑛 ⨀𝐾1,2 is a Super root square mean graph.
Example 2.4: Super root square mean labeling of 𝑃4 ⨀𝐾1,2 is shown below.
Figure 2
Theorem 2.5: 𝑃𝑛 ⨀𝐾1,3 is a super root square mean graph.
Proof: Let 𝑢1 , 𝑢2 , … , 𝑢𝑛 be the path 𝑃𝑛 . Let 𝑣𝑖 ,𝑤𝑖 and 𝑠𝑖 , 1 ≤ 𝑖 ≤ 𝑛 be the vertices of 𝐾1,3 attached to 𝑢𝑖 .
Define a function 𝑓: 𝑉(𝑃𝑛 ⨀𝐾1,3 ) → 1,2, … , 𝑝 + 𝑞 by
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Vol: I. Issue XXXV, January 2016
ISSN: 2320-5504, E-ISSN-2347-4793
𝑓 𝑢𝑖 = 8𝑖 − 5 , 1 ≤ 𝑖 ≤ n
𝑓 𝑣𝑖 = 8𝑖 − 7, 1 ≤ 𝑖 ≤ n
𝑓 𝑤𝑖 = 8𝑖 − 3 , 1 ≤ 𝑖 ≤ n
𝑓 𝑠𝑖 = 8𝑖 − 1 , 1 ≤ 𝑖 ≤ n
Then the edges are labeled as
𝑓 𝑢𝑖 𝑢𝑖+1 = 8𝑖, 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑣𝑖 = 8𝑖 − 6 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑢𝑖 𝑤𝑖 = 8𝑖 − 4 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑢𝑖 𝑠𝑖 = 8𝑖 − 2 , 1 ≤ 𝑖 ≤ 𝑛
Then 𝑓 𝑉 ∪ 𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
= 1,2, … , 𝑝 + 𝑞 . Hence by definition 1.1, 𝑃𝑛 ⨀𝐾1,3 is a Super root square mean graph.
Example 2.6: Super root square mean labeling of 𝑃4 ⨀𝐾1,3 is shown below.
Figure 3
Theorem 2.7: 𝑇𝑛 ⨀𝐾1 is a Super Root Square Mean graph.
Proof: Let 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 be a path of length 𝑛. Let 𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1 be the new vertex joined to 𝑢𝑖 and 𝑢𝑖+1 .The
resulting graph is called 𝑇𝑛 . Let 𝑥𝑖 be the vertex which is joined to 𝑢𝑖 , 1 ≤ 𝑖 ≤ 𝑛. Let 𝑦𝑖 be the vertex which is joined to
𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1. The resulting graph is 𝑇𝑛 ⨀𝐾1 . Let 𝐺 = 𝑇𝑛 ⨀𝐾1 .
Define a function 𝑓: 𝑉 𝐺 → {1,2, … , 𝑝 + 𝑞} by
𝑓 𝑢𝑖 = 9𝑖 − 6 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑣𝑖 = 9𝑖 − 4 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑥𝑖 = 9𝑖 − 38 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑦𝑖 = 9𝑖 − 2 , 1 ≤ 𝑖 ≤ 𝑛 − 1
Then the edges are labeled as
𝑓 𝑢𝑖 𝑢𝑖+1 = 9𝑖 − 1 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑣𝑖 = 9𝑖 − 5 , 1 ≤ 𝑖 ≤ 𝑛 − 1
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𝑓 𝑣𝑖 𝑢𝑖+1 = 9𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑥𝑖 = 9𝑖 − 7 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑣𝑖 𝑦𝑖 = 9𝑖 − 3 , 1 ≤ 𝑖 ≤ 𝑛 − 1
Then 𝑓 𝑉 ∪ 𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
= 1,2, … , 𝑝 + 𝑞 . Hence by definition 1.1, 𝑇𝑛 ⨀𝐾1 is a Super Root Square Mean graph.
Example2.8: The Super Root Square Mean labeling of 𝑇4 ⨀𝐾1 is given below.
Figure 4
Theorem2. 9: 𝑄𝑛 ⨀𝐾1 is a Root Square Mean graph.
Proof: Let 𝑢1, 𝑢2 , ⋯ , 𝑢𝑛 be a path. Let 𝑣𝑖 and 𝑤𝑖 be two vertices joined to 𝑢𝑖 and 𝑢𝑖+1 respectively and then join 𝑣𝑖 and
𝑤𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1. The resulting graph is called as quadrilateral snake 𝑄𝑛 . Let 𝑥𝑖 be the new vertex joined to 𝑢𝑖 , 1 ≤
𝑖 ≤ 𝑛. Let 𝑦𝑖 be the new vertex joined to 𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1. Let 𝑧𝑖 be the new vertex joined to 𝑤𝑖 , 1 ≤ 𝑖 ≤ 𝑛 − 1. The
resulting graph is 𝑄𝑛 ⨀𝐾1 . Let 𝐺 = 𝑄𝑛 ⨀𝐾1 .
Define a function 𝑓: 𝑉 𝐺 → {1,2, … , 𝑝 + 𝑞} by
𝑓 𝑢𝑖 = 13𝑖 − 10 , 1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑣𝑖 = 13𝑖 − 8 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑤1 = 11,
𝑓 𝑥1 = 1,
𝑓 𝑤𝑖 = 13𝑖 − 1 , 2 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑥𝑖 = 13𝑖 − 13 , 2 ≤ 𝑖 ≤ 𝑛
𝑓 𝑦𝑖 = 13𝑖 − 6 , 1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑧𝑖 = 13𝑖 − 4 , 1 ≤ 𝑖 ≤ 𝑛 − 1
Then the edges are labeled as
𝑓 𝑢1 𝑢2 = 12,
𝑓 𝑢𝑖 𝑢𝑖+1 = 13𝑖 − 2 ,2 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑢𝑖 𝑥𝑖 = 13𝑖 − 11 ,1 ≤ 𝑖 ≤ 𝑛
𝑓 𝑢𝑖 𝑣𝑖 = 13𝑖 − 9 ,1 ≤ 𝑖 ≤ 𝑛 − 1
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𝑓 𝑢𝑖+1 𝑤𝑖 = 13𝑖 + 1 ,1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑣𝑖 𝑤𝑖 = 13𝑖 − 5,1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑣𝑖 𝑦𝑖 = 13𝑖 − 7 ,1 ≤ 𝑖 ≤ 𝑛 − 1
𝑓 𝑤𝑖 𝑧𝑖 = 13𝑖 − 3 ,1 ≤ 𝑖 ≤ 𝑛 − 1
Then 𝑓 𝑉 ∪ 𝑓 ∗ 𝑒 : 𝑒 ∈ 𝐸 𝐺
= 1,2, … , 𝑝 + 𝑞 . Hence by definition 1.1, 𝑄𝑛 ⨀𝐾1 is a Super Root Square Mean graph.
Example2.10: The labeling pattern of 𝑄4 ⨀𝐾1 is given below.
Figure 5
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