SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v14i35.1336Research ArticleSuper Root Cube of Cube Difference Labeling of Some Special GraphsVembarasiGvembagopal@gmail.com1GowriR2Research Scholar, Department of Mathematics, Government College for Women (Autonomous),KumbakonamIndiaAssistant professor, Department of Mathematics, Government College for Women (Autonomous)KumbakonamIndia143527782021Abstract

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.

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.

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 f*(e=uv)=f(u)2+f(v)22 (or) f(u)2+f(v)22 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 mean labeling is called a super root mean graph.

Let G=(V(G),E(G)) be a graph. G is said to be a cube difference labeling if there exists a injective function f:v(G)→{0,1,…,p-1} such that the induced function f*:E(G)→N is given by f*(uv)=[f(u)]3-[f(v)]3 is injective.

Let G=(V(G),E(G)) be a graph. G is said to be a cube difference labeling if there exists a injective function f:v(G)→{0,1,…,p-1} such that the induced function f*:E(G)→N is given by f*(uv)=∣[f(u)]3-f(v)3∣ (or) [f(u)]3-[f(v)]3 is injective.

Super Root Cube Of Cube Difference Labeling of GraphsDefinition 3.1

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 f*(e=uv)=f(u)3-f(v)313 (or) f(u)3-f(v)313 , then f is called a super root cube of cube difference labeling if f(v)∪f*(e):e∈E(G)={1,2,…,p+q} . A graph is called a super root cube of cube difference labeling.

Theorem 3.2

Triangular Snake Tn is a super root cube of cube difference labeling of graph.

Proof : 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 That is every edge of a path is replaced by a triangular C3 . Define the function f:V(G)→{1,2,…,p+q} by

fui=2i, for 1≤i≤nfvi=2i-1, for 1≤i≤n.

And the induced edge labeling function f*:E(G)→N defined by

f*(e=uv)=f(u)3-f(v)313

Then the edge sets are,

f*uiui+1=24i2+24i+813, for 1≤i≤n-1

f*un-1un=24n2-24n+813

f*uivi=12i2-6i+113, for 1≤i≤n

f*unvn=12n2-6n+113

f*ui+1vi=∣36i2+18i+913∣ for 1≤i≤n

f*un+1vn=36n2+90n+6313

Hence the graph G is a Super root cube of cube difference labeling.

Example 3.3

Super root cube of cube difference labeling of T4 is given below.

The Cycle graph Cn is a Super root cube of cube difference labeling.

Proof : A closed path is called a cycle. A cycle on n vertices is denoted by Cn graph with vertices u1,u2,…,un and the edges e1,e2,…,em Define the function f:V(G)→{1,2,…,p+q} by

fui=ifor1≤i≤n

And the induced edge labeling function f*:E(G)→N defined by

f*(e=uv)=f(u)3-f(v)313

Then the edges labels are,

f*uiui+1=3i2+3i+113, for 1≤i≤n-1

f*un-1un=3n2-3n+113

f*unu1=n3-113

Hence the graph Cn is a Super root cube of cube difference labeling.

Example 3.5

The following is an example for C7 is a Super root cube of cube difference labeling of graph.

Let G be a graph obtained by attaching a pendant edge to both sides of each vertex of a path Pn. Then G is a Super root cube of cube difference labeling of graph only if n≥5. .

Proof: Let G be path Pn. The graph obtained by attaching pendant edges to both sides of each vertex. Let xi,yi and zi for 1≤i≤n be the new vertices of G. Define the function f:V(G)→{1,2,…,p+q} by

fxi=3i-1for1≤i≤nfyi=3ifor1≤i≤nfzi=3i-2for1≤i≤n

And the induced edge labeling function f*:E(G)→N defined by

f*(e=uv)=f(u)3-f(v)313∣1

Then the edge sets are,

f*xixi+1=∣81i2+27i+913∣, for 1≤i≤n-1

f*xiyi=27i2-9i+113, for 1≤i≤n

f*xizi=27i2-27i+713, for 1≤i≤n

Hence the graph G is a Super root cube of cube difference labeling.

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.

ReferencesShiamaJ.Square Difference Labeling for Some GraphsShiamaJCube Difference Labeling of Some GraphsSandhyaS. S.SomasundaramS.AnusaS.Root square mean labeling of graphsKannanMPrasadRGopiRSuper Root Square Mean Labeling of Disconnected GraphsGowriR.VembarasiG.Root Cube Mean Labeling of GraphsGowriR.VembarasiG.Root Cube Mean Labeling of GraphsThereseS.KulandhaiKRomilaCube Root Cube Mean Labeling of GraphsGowriRVembarasiGRoot Cube of Cube Difference Labeling of GraphsThiruganasambandamKVenkatesanKSuper root square mean labeling of graphs