# On Super Mean Labeling for Total Graph of Path and Cycle

Authors: Nur Inayah; I. Wayan Sudarsana; Selvy Musdalifah; Nurhasanah Daeng Mangesa
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

LetG(V,E) be a graph with the vertex set V and the edge set E, respectively. By a graph G=(V,E) we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of G is called order of G and it is denoted by p. Let G be a (p,q) graph. A super mean graph on G is an injection f:V→{1,2,3…,p+q} such that, for each edge e=uv in E labeled by f⁎e=fu+f(v)/2, the set fV∪{f⁎e:e∈E} forms 1,2,3,…,p+q. A graph which admits super mean labeling is called super mean graph. Thetotal graphT(G)ofG is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G. We have showed that graphs T(Pn) and TCn are super mean, where Pn is a path on n vertices and Cn is a cycle on n vertices.


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## Body

## 1. Introduction and Preliminary Results

LetG(V,E) be a graph with the vertex set V and the edge set E, respectively. By a graph G=(V,E) we mean a finite undirected, graph with neither loops nor multiple edges. The number of vertices of G is called order of G and it is denoted by p. The number of edges of G is called size of G and it is denoted by q. A (p,q) graph G is a graph with p vertices and q edges. Terms and notations not defined here are used in the sense of Harary [1].In 2003, Somasundaram and Ponraj [2] have introduced the notion of mean labelings of graphs. Let G be a (p,q) graph. A graph G is called a mean graph if there is an injective function f from the vertices ofGto {0,1,2,···,q} such that when each edge e=uv is labeled with f∗(e)=(fu+fv+1)/2 if fu+fv is even and f∗e=fu+fv+1/2 if fu+f(v) is odd, then the resulting edge labels are distinct. Furthermore, the concept of super mean labeling was introduced by Ponraj and Ramya [3]. Let f:V→{1,2,3,···,p+q} be an injection on G. For each edge e=uv and an integer m≥2, the induced Smarandachely edge m-labelingf∗ is defined by f∗e=uv=fu+f(v)/m. Then f is called a Smarandachely super m-mean labeling if f(V)∪{f∗(e):e∈E}={1,2,3,···,p+q}. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph. Particularly, if m=2, we know that(1)f∗e=uv=fu+fv2,iffu+fviseven;fu+fv+12,iffu+fvisodd.Such a labelingf is called a super mean labeling of G if f(V)∪{f∗(e):e∈E}={1,2,3,···,p+q}. A graph that admits a super mean labeling is called a super mean graph. Further discussions of mean and super mean labelings for some families of graph are provided in [4–10] and Gallian [11].Thetotal graphT(G) of G is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G. For instance, when G=Pn, total graph of path T(Pn) is provided in Figure 1. Since the problem on super mean labeling for total graph of path and cycle are still open, the new our contributions are stated in the following sections.Figure 1
The total graph of path onnvertices.


## 2. On Super Mean Labeling for Total Graph of Path

The theorem proposed in this section deals with the super mean labeling for total graph of path onn vertices, T(Pn).Theorem 1.
The total graph of path on n vertices,T(Pn), is a super mean graph for all n≥3.Proof.
LetVTPn=vi:1≤i≤n∪{ui:1≤i≤n-1} and ETPn=ei,ei′,ei′′:1≤i≤n-1∪eim:1≤i≤n-2 with ei=vivi+1, ei′=viui, ei′′=vi+1ui for 1≤i≤n–1, and ei′′′=uiui+1 for 1≤i≤n–2. Immediately, we have that the cardinality of the vertex set and the edge set of T(Pn) are p=2n-1 and q=4n-5, respectively, and so p+q=6n-6. Define an injection f:VTPn→{1,2,…,6n-6} for n≥3 as follows. fvi=2i-1 for i=1,2,…,n. fui=2i+4n-4 for i=1,2,…,n-1.And so we have(2)f∗ei=2ifori=1,2,…,n-1.f∗ei′=2n+2i-2fori=1,2,…,n-1.f∗ei′′=2n+2i-1fori=1,2,…,n-1.f∗ei′′′=4n+2i-3fori=1,2,…,n-2.Next, we consider the following sets: (3)A1=fvi=2i-1:i=1,2,…,n;A2=fui=4n+2i-4:i=1,2,…,n-1;A3=f∗ei=2i:i=1,2,…,n-1;A4=f∗ei′=2n+2i-2:i=1,2,…,n-1;A5=f∗ei′′=2n+2i-1:i=1,2,…,n-1;A6=f∗ei′′′=4n+2i-3:i=1,2,…,n-2. It can be verified that fVTPn∪f∗ETPn=⋃i=16Ai={1,2,3,…,6n-6} and so f is a super mean labeling of TPn. Hence, TPn is a super mean graph. For a simple example, the super mean labeling for total graph of path on five vertices is provided in Figure 2.Figure 2
The super mean labeling for total graph of path on 5 vertices,T(P5).


## 3. On Super Mean Labeling for Total Graph of Cycle

The theorem proposed in this section deals with the super mean labeling for total graph of cycle onn vertices, T(Cn). For illustration, total graph of cycle on n vertices is provided in Figure 3.Figure 3
The total graph of cycle onnvertices, T(Cn).Theorem 2.
The total graph of cycle on n vertices,TCn, is a super mean graph if either n is odd and n≥3 or n is even and n≥6.Proof.
LetVTCn=vi,ui:1≤i≤n and ETCn=ei,ei′,ei′′,ei′′′:1≤i≤n, where (4)ei=vivi+1,1≤i≤n-1;v1vi,i=n.ei′=viuifor1≤i≤nei′′=vi+1ui,1≤i≤n-1;v1ui,i=n.ei′′′=uiui+1,1≤i≤n-1;u1ui,i=n.Immediately, we have that the cardinality of the vertex set and the edge set of T(Cn) are p=2n and q=4n, respectively, and so p+q=6n.Define an injectionf:VT(Cn)→{1,2,…,6n} for odd n≥3 as follows:(5)fvi=2i-1i=1,2,3,…,n2;2ii=n2+1,n2+2,…,n.fui=4n+2i-1,i=1,2,3,…,n2;4n+2i,i=n2+1,n2+2,…,n. And so we have(6)f∗ei=2i,i=1,2,3,…,n2-1;n+2,i=n2;2i+1,i=n2+1,n2+2,…,n-1;n+1,i=n.fei′=2n+2i-1,i=1,2,3,…,n2;2n+2i,i=n2+1,n2+2,…,n.f∗ei′′=2n+2i,i=1,2,3,…,n2-1;3n+2,i=n2;2n+2i+1,i=n2+1,n2+2,…,n-1;3n+1,i=n.f∗ei′′′=4n+2i,i=1,2,3,…,n2-1;5n+2,i=n2;4n+2i+1,i=n2+1,n2+2,…,n-1;5n+1,i=n.Next, we consider the following sets:(7)A1=fvi=2i-1:i=1,2,…,n2;A2=fvi=2i:i=n2+1,n2+2,…,n;A3=fui=4n+2i-1:i=1,2,…,n2;A4=fui=4n+2i:i=n2+1,n2+2,…,n;A5=f∗ei=2i:i=1,2,…,n2-1;A6=f∗ei=n+2:i=n2;A7=f∗ei=2i+1:i=n2+1,n2+2,…,n-1;A8=fei=n+1:i=n;A9=f∗ei′=2n+2i-1:i=1,2,…,n2;A10=f∗ei′=2n+2i:i=n2+1,n2+2,…,n;A11=f∗ei′′=2n+2i:i=1,2,…,n2-1;A12=f∗ei′′=3n+2:i=n2;A13=f∗ei′′=2n+2i+1:i=n2+1,n2+2,…,n-1;A14=f∗ei′′=3n+1:i=n;A15=f∗ei′′′=4n+2i:i=1,2,…,n2-1;A16=f∗ei′′′=5n+2:i=n2;A17=f∗ei′′′=4n+2i+1:i=n2+1,n2+2,…,n-1;A18=f∗ei′′′=5n+1:i=n.It can be verified that fVTCn∪f∗ETCn=⋃i=118Ai=1,2,3,…,6n and so f is a super mean labeling of TCn. Hence TCn is a super mean graph for odd n≥3.Now define an injectionf1:VTCn→1,2,3,…,6n for even n≥6 as follows:(8)f1vi=1,i=1;3i-3,i=2,3;4i-7,i=4,5,6,…n2+1;4n-4i+8,i=n2+2,n3+3,…,n-17,i=nf1ui=4n+1,i=1;4n+3i-3,i=2,3;4n+4i-7,i=4,5,6,…n2+1;8n-4i+8,i=n2+2,n3+3,…,n-14n+7,i=nand so we have (9)f1∗ei=2,i=1;3i-1,i=2,3;4i-5,i=4,5,6,…n2+1;4n-4i+6,i=n2+2,n3+3,…,n-14,i=nf1∗ei′=2n+1,i=1;2n+3i-3,i=2,3;2n+4i-7,i=4,5,6,…n2+1;6n-4i+8,i=n2+2,n3+3,…,n-12n+7,i=nf1∗ei′′=2n+2,i=1;2n+3i-1,i=2,3;2n+4i-5,i=4,5,6,…n2+1;6n-4i+6,i=n2+2,n3+3,…,n-12n+4,i=nf1∗ei′′′=4n+2,i=1;4n+3i-1,i=2,3;4n+4i-5,i=4,5,6,…n2+1;8n-4i+6,i=n2+2,n3+3,…,n-14n+4,i=nIt can be verified that f1VTCn∪f1∗ETCn=1,2,3,…,6n and sof1 is a super mean labeling of TCn. Hence TCn is a super mean graph for even n≥6. For illustration, a super mean labeling for total graph of cycle on8 vertices is provided in Figure 4.Figure 4
The super mean labeling for total graph of cycle on 8 vertices,T(C8).


## 4. The Duality of Super Mean Labeling

LetG be a (p,q) graph. Given any Smarandachely super 2-mean labeling λ on graph G, the labeling λ1 defined by(10)λ1x=p+q+1-λx,foranyvertexx,λ1∗xy=p+q+1-λ∗xy,foranyedgesxyis also a Smarandachely super 2-mean labeling of G.For the proof, sinceλ is an injection then it is follows that λ1 is also an injection on G. Hence it can be verified that the set λ1(V)∪{λ∗1(e):e∈E} forms {1,2,3,···,p+q} and so the injection λ1 is also a Smarandachely super 2-mean labeling on graph G. Furthermore we call that the labeling λ1 is adualsuper mean labeling of λ.By using the duality property above, Theorems1 and 2, we have the following corollary.Corollary 3.
LetTPn and TCn be the total graph of path and cycle with n vertices, respectively.(i)
For alln≥3, if λvi=6n-2i-4 for 1≤i≤n and λui=2n-2i-1 for 1≤i≤n-1 then λ is a super mean labeling for T(Pn).(ii)
For oddn≥3 if(11)λvi=6n-2i+2,i=1,2,3,…,n2;6n-2i+1i=n2+1,n2+2,…,n,λui=2n-2i+2,i=1,2,3,…,n2;2n-2i+1i=n2+1,n2+2,…,nthenλ is a super mean labeling for TCn.(iii)
For evenn≥6, if(12)λ1vi=6n,i=1;6n-3i+4,i=2,3;6n-4i+8,i=4,5,6,…n2+1;2n+4i-7,i=n2+2,n3+3,…,n-16n-6,i=n,λ1ui=2n,i=1;2n-3i+4,i=2,3;2n-4i+8,i=4,5,6,…n2+1;2n-4i-7,i=n2+2,n3+3,…,n-12n-6,i=nthenλ1 is a super mean labeling for TCn.


## 5. Summary and Remarks

Here we propose new results corresponding to super mean labeling for total graph of path and cycle. This work is an effort to relate Smarandachely superm-mean labeling and its dual for m≥2. All results reported here are in total graph of path and cycle, T(Pn) and T(Cn). In future, it is not only possible to investigate some more results corresponding to other graph families but also Smarandachely superm-mean labeling in general as well.



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