Super Root Cube of Cube Diﬀerence Labeling of Some Special Graphs

Background/Objectives: This study gives an extended and the new kinds of super root cube of cube diﬀerence labeling of some graphs are obtained. Methods/ Findings: We derive super root cube of cube diﬀerence labeling of path related graph and analyzed cycle related graphs.


Introduction
All graphs G = (V (G), E(G)) with p vertices and q edges we mean a simple connected and undirected graph. In 2012, J. Shiama (1) , studied square difference labeling of some graphs. In 2013, J. Shiama (2) , introduced the concept of cube difference labelings and investigated the labelings for certain graphs. S.Sandhya et.al (3) , was initiated the concept of root square mean labeling of graphs. In 2016, M. Kannan et.al (4) , introduced the concept of super root square mean labeling of disconnected graphs are discussed. In 2017, R.Gowri and G.Vembarasi (5) , was discussed root cube mean labeling of graphs. R.Gowri and G.Vembarasi (6) , extended the new concept of root cube difference labeling of graphs are introduced in 2018. In 2019, S.Kulandhai Theresa and K.Romila (7) , was discussed the concept of cube root cube mean labeling of graphs are introduced. In 2020, R.Gowri and G.Vembarasi (8) recently introduced the concept of root cube of cube difference labeling of graphs. Likewise, many authors have discussed this topic in their work. In this study we discuss about the super root cube of cube difference labeling and investigate certain families of graphs.

Definition 2.3 (5)
The product graph P 2 × P n is called a Ladder and it is denoted by L n .

Definition 2.4 (9)
A graph G with p vertices and q edges then f : V (G) → {1, 2, . . . , p+q} be an injective function. For each edge e=uv . Let A graph that admits a super root mean labeling is called a super root mean graph.

Definition 2.5 (2)
Let G = (V (G), E(G)) be a graph. G is said to be a cube difference labeling if there exists a injective function

Definition 2.6 (6)
Let G = (V (G), E(G)) be a graph. G is said to be a cube difference labeling if there exists a injective func-

Super Root Cube Of Cube Difference Labeling of Graphs Definition 3.1
A graph G with p vertices and q edges then f : ] , then f is called a super root cube of cube difference labeling A graph is called a super root cube of cube difference labeling.

Theorem 3.2
Triangular Snake T n is a super root cube of cube difference labeling of graph.
Proof : A Triangular Snake T n is obtained from a path u 1 , u 2 , . . . , u n by joining u i and u i+1 to a new vertex v i for 1 ≤ i ≤ n That is every edge of a path is replaced by a triangular C 3 . Define the function f : And the induced edge labeling function f * : E(G) → N defined by Then the edge sets are, Hence the graph G is a Super root cube of cube difference labeling.

Example 3.3
Super root cube of cube difference labeling of T 4 is given below.

Theorem 3.4
The Cycle graph C n is a Super root cube of cube difference labeling.
Then the edges labels are, Hence the graph C n is a Super root cube of cube difference labeling. https://www.indjst.org/

Example 3.5
The following is an example for C 7 is a Super root cube of cube difference labeling of graph.

Theorem 3.6
The crown C n ΘK 1 is a Super root cube of cube difference labeling of graph.
Proof: Let C n be the Cycle u 1 u 2 ..u n u 1 and v i be the pendant vertices adjacent to u i , 1 ≤ i ≤ n . Define the function f : V (C n ΘK 1 ) → {1, 2, . . . , p + q} by And the induced edge labeling function f * : E(G) → N defined by Then the edge sets are, 1 3 Hence the graph G is a Super root cube of cube difference labeling. https://www.indjst.org/

Example 3.7
Super root cube of cube difference labeling of C n ΘK 1 is given below.
And the induced edge labeling function f * : E(G) → N defined by Then the edge sets are, Hence the graph G is a Super root cube of cube difference labeling.

Conclusion
In this article we discussed the concept of Super Root Cube of Cube Difference Labeling of Graphs are initiated and also some graphs are introduced and characterized. Then the relative results between path, cycle related graphs are discussed. Here all the edge values are distinct and the resulting edge values do not exceed the vertex value.