Some More Results on Root Square Mean Graphs

A graph G = (V, E) with p vertices and q edges is called a Root Square Mean graph if it is possible to label the vertices x ∈ V with distinct elements f(x) from 1,2, ... , q + 1 in such a way that when each edge e = uv is labeled with f(e = uv) = ⌈√ f(u)2+f(v)2 2 ⌉ or ⌊√ f(u)2+f(v)2 2 ⌋ , then the resulting edge labels are distinct. In this case f is called a Root Square Mean labeling of G. The concept of Root Square Mean labeling was introduced by (S. S. Sandhya, S. Somasundaram and S. Anusa). We investigated the Root Square Mean labeling of several standard graphs such as Path, Cycle, Comb, Ladder, Triangular snake, Quadrilateral snake etc., In this paper, we investigate the Root Square Mean labeling for Double Triangular snake, Alternate Double Triangular snake, Double Quadrilateral snake, Alternate Double Quadrilateral snake, and Polygonal chain.


Introduction
The graph considered here will be finite, undirected and simple.The vertex set is denoted by () and the edge set is denoted by ().For all detailed survey of graph labeling we refer to Gallian (2010).For all other standard terminology and notations we follow Harary (1988).A Triangular snake   is obtained from a path  1  2  3 ⋯   by joining   and  +1 to a new vertex   for 1 ≤  ≤  − 1.A Double Triangular Snake (  ) consists of two Triangular snakes that have a common path.An Alternate Triangular snake (  ) is obtained from a path  1  2 …   by joining   and  +1 (Alternatively) to new vertex   .An Alternate Double Triangular Snake ((  )) consists of two Alternate Triangular snakes that have a common path.A Quadrilateral snake   is obtained from a path  1  2 …   by joining   and  +1 to new vertices   and   respectively and then joining   and   .A Double Quadrilateral snake (  ) consists of two Quadrilateral snakes that have a common path.An Alternate Quadrilateral snake (  ) is obtained from a path  1  2 …   by joining   and  +1 (Alternatively) to new vertices   and   respectively and then joining   and   .An Alternate Double Quadrilateral snake ((  )) consists of two Alternate Quadrilateral snakes that have a common path.A Polygonal chain  , is a connected graph all of whose  blocks are polygons on  sides.S. Somasundaram and R. Ponraj introduced the concept of mean labeling of graphs and investigated the mean labeling of some standard graphs.S. Somasundaram and S. S. Sandhya introduced the concept of Harmonic mean labeling of graphs.S. ⌋ , then the edge labels are distinct.In this case  is called Harmonic mean labeling of .
Definition 2.3: A graph with  vertices and  edges is called as Geometric mean graph if it is possible to label the vertices  ∈  with distinct elements () from 1,2,3, … ,  + 1 in such a way that when each edge  =  is labeled with ( = ) = ⌈√()() ⌉  ⌊√()() ⌋ , then the edge labels are distinct.In this case  is called Geometric mean labeling of .
Definition 2.4: A graph with  vertices and  edges is called a Root Square mean graph if it is possible to label the vertices  ∈  with distinct elements () from 1,2,3, … ,  + 1 in such a way that when each edge ⌋ , then the edge labels are distinct.In this case  is called Root Square mean labeling of .
Theorem 2.5: Double Triangular snakes (  ) are Root square mean graphs.
The edges are labeled as Then the edge labels are distinct.Hence Double Triangular snakes are Root Square mean graphs.
Example 2.6: Root Square mean labeling of ( 5 ) is given below.
The edges are labeled as Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of .
The labeling pattern is shown below.The edges are labeled as Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of .
The labeling pattern is shown below.
The edges are labeled as Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of  .
The labeling pattern is shown below.Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 5 From all the above cases, we conclude that Alternate Double Triangular Snakes ((  )) are Root Square mean graphs.
The edges are labeled as Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of .
Example 2.9: The Root Square mean labeling of ( 5 ) is given below.
Case 1: If the Double Quadrilateral starts from  1 , then we consider two sub cases.

Sub Case 1(a):
The edges are labeled as Then the edge labels are distinct.Hence in this case  is a Root Square mean labeling of .
The labeling pattern is shown below.Hence the graph  , has distinct edge labels, hence  , is a Root Square mean graph.
Example 2.12: Root Square mean labeling of  4,6 chain is given below.

Figure 2 Sub
Figure 2

Figure 4 Sub
Figure 4

Figure 7 Sub
Figure 7

Figure 9 Sub
Figure 9

Figure 10
Figure 10 Theorem 2.11: Polygonal chain  , are Root Square mean graphs for all  and .

2. Root Square Mean Labeling Definition 2.1: A
Somasundaram and P. Vidhya Rani introduced the concept of Geometric mean labeling of graphs.In this paper we prove that Double Triangular snake, Alternate Double Triangular snake, Double Quadrilateral snake, Alternate Double Quadrilateral snake and Polygonal Chains are Root Square mean graphs.(S. S. Sandhya, S. Somasundaram & S. Anusa) Double Triangular and Double Quadrilateral snakes are mean graphs.(S. S. Sandhya & S. Somasundaram) Triangular snakes and Quadrilateral snakes are Harmonic mean graphs.(C.Jaya Sekaran, S. S. Sandhya & C. David Raj) Double Triangular snakes and Alternate Double Triangular snakes are Harmonic mean graphs.graph  = (, ) with  vertices and  edges is called a mean graph if it is possible to label the vertices  ∈  with distinct elements () from 0,1,2, … ,  in such a way that when each edge Theorem 2.8: Double Quadrilateral snake graph (  ) is a Root Square mean graphs.Let   be the path  1  2  3 ⋯   .To construct (  ), join   and  +1 to four new vertices   ,   ,   ′ and   ′ by the edges     ,  +1   ,     ,     ′ ,  +1   ′ and   ′   ′ , for 1 ≤  ≤  − 1. Proof: