Python Programming on Fibonacci Graceful Anti-Magic Labeling for Star Graphs

Abstract

A graph G=(V,E,?) with p vertices and q edges is called Fibonacci Graceful Anti magic Graph [FGAMG] if a function ? is defined as ?:V?{0,1,2,…,F_(q+1) } and the induced edge labeling ?^*:E?{F_2,F_(3,) F_4,…,F_(q+1) } is defined by ?^* (uv)=|?(u)-?(v)|is bijective. In addition all the vertex sums are pairwise distinct and all the edges are unique. In this article the concept of Fibonacci Graceful Anti- Magic Labeling [FGAML] is introduce and investigate some star related graphs which are Fibonacci Graceful Anti magic Graph [FGAMG]. AMS Classification : 05C78

Introduction

I. INTRODUCTION

Fibonacci graceful labeling idea was  introduced by E.Barkaukas et.all [3]. Hartsfield and Ringel introduced the concept of Anti-Magic labeling. The concept of Fibonacci mean Anti-magic labeling was introduced by Ameenalbibi and T.Ranjani[1]. Now this article is based on connected, undirected and Star related Graphs. Python is the most popular language in world wide. It was introduced  by GuioVan Rossum on Feburary 20,1991.Fibonacci Graceful Anti- Magic Labeling FGAML concept is introduce here. While  investigating  some star based graphs  it is  found that they are  Fibonacci Graceful Anti magic Graph FGAMG. Here Python coding is generated for  the Fibonacci Graceful Anti-Magic Labeling.

II. DEFINITIONS

1. Definition 2.1: The Fibonacci numbers F1,F2,F3,…  are defined by F0=0,F1=1,F2=1,… and Fn+1=Fn+Fn-1 , n≥1. The Fibonacci sequence is 1,1,2,3,5…
2. Definition 2.2: The function  ?:V(G)?{0,1,2,…,Fq+1}   (here Fq  is the  qth  Fibonacci number) is called  FGAML   if the induced edge labeling  ?* : E(G)?{F2,F3,F4,…,Fq+1}  defined by  ?*(uv)=?(u)-?(v)  is bijective. In addition all the vertex sums are pairwise distinct and all the edges are unique.
3. Definition 2.3 : Crown  graph is  obtained  by  joining  a single pendant  edge to each vertex of the cycle Cn . It is denoted byCn?K1 .
4. Definition 2.4: Double star K1,n,n   is a tree obtained from the star K1,n  by adding a new pendant edge to each  of  the existing  n  pendant vertices.
5. Definition2.5 : A Bi-star Bn,n  graph  is obtained  by  adding  the central vertices of two copies of  Star.
6. Definition 2.6 : A Bi-Shell BSn,n graph isobtained  by adding  the central vertices of two copies of  Shell.
7. Definition 2.7 : The Vertex sum at one vertex is the sum of labels of all edges incident to such a           vertex.

III. RESULTS

1) Theorem 

Crown graph  Cn?K1  is FGAMG   for all n .

Proof:

Let Cn?K1  be the crown. The order of the crown is p=2n  and the sizeis  q=2n.    Let  VG={a1,a2,…an,b1,b2,…,bn } here  a1,a2,…,an   are  the  vertices  of  the cycle Cn  and bi 's  are  the pendant  vertices  adjacent to ai's  of the cycle Cn . The edge set  EG=eii,eij  where eii=(aibi)  and eij=aiaj.

Define ?:V?{0,1,2,…,Fq+1}

 ?a1=0                                                               

?(ai+1) = F2i+2       for i= 1,2,3,…,n-1

?(bi+1) = F2i+1      for i=1, 3…..n-1

?b3=6

 ?b1=F2n+1                                                     

The edge labels are

eii=?*(aibi)=?ai-?bi           for  i=1,2,3,…,n

eij=?*(aiaj)=?ai-?aj           for  i=1,2,3,…,n and for  j=i+1

Here all the vertex sums are pairwise distinct and all the edges are unique.

Thus  ?  admitsFGAML .

Hence Crown  Cn?K1   is  FGAMG . 

Python coding and output is enclosed in the Appendix .

Illustration 3.1

Figure 1. The Crown C6?K1  is FGAMG

2) Theorem

Double star graph is  FGAMG   for all n.

Proof :

Let  K1,n,n  be the double star. The order of K1,n,n is p=2n+1  and the size q=2n. By the definition of  K1,n,n  the vertex set V(G) = a, b1, b2 ,…, bn,   c1,c2,…,cn .Let a  be the central vertex, and b1, b2 ,…,bn  be the vertices of the star and c1,c2,…,cn  be the vertices adding new pendent edge to each existing of the n  pendent edges.The edge set  EG=ei,eii here ei=(abi)  and eii=bici.

Define  ?:V?{0,1,2…..Fq+1}  as follows

? a=0,

? bi=F2n+2-i                      i =  1, 2, . . .,n .

 ?ci=? bi-Fn+2-i           i= 1,2,…,n .

The edge labels are

eii=?*(bici)=?bi-?ci           for  i=1,2,3,…,n

ei=?*(abi)=?a-?bi              for  i=1,2,3,…,n

Here all the vertex  sums are pairwise distinct and all the edges are  unique.

Thus ?  admits AML  .

Hence the double star K1,n,n  isFGAMG .

Python coding and output is enclosed in the  Appendix .

Illustration 3.2

Fig 2. Double star K1,4,4  is FGAMG

3) Theorem

Bi-star  Bn,n  is FGAMG  for all n.

Proof:

Let Bn,n  be Bi-star, By the definition ofBn,n , The order of the Bi-star  Bn,n is p=2n+2 and size is q=2n+1 .The vertex set VG={a0,a1,a2,…,an,b0,b1,b2,…,bn} . Let a1,a2,…,an  be the adjacent vertices of a0  and b1,b2,…,bn  be the adjacent vertices of b0 .

EG=e1*,ej,ej hereei=b0bi,gi=a0ai,e1*=(a0b0) .

Define ?:V?{0,1,2,…,Fq+1}  as follows

?a0=0                                                                

?ai+1= Fn+2+i                 for i= 1,2,3,…,n-1

?bi+1= Fi+2+F2           for i=1, 2,3,…,n-1

?b0=F2

The edge labels are

ei=?*(b0bi)=?b0-?bi           for  i=1,2,3,…,n

gi=?*(a0ai)=?a0-?ai           for  i=1,2,3,…,n

e1*=?*(a0b0)=?a0-?b0          

Here all the vertex  sums are pairwise distinct and all the edges are  unique.

Thus  ?  admits  FGAML .

Hence the graph Bn,n is FGAMG .

Python coding and output is enclosed in  the Appendix .

Illustration 3.3

Fig 3. Bi-starB5,5  is FGAMG

4) Theorem

 Bi-shell BSn,n  is FGAMG  for all n.

Proof:

Let BSn,n  be the Bi-shell graph, The order of  Bn,n is p=2n+2 and the size is q=4n-1 . The vertex set VG=a0,a1,a2,…,an,b0,b1,b2,…,bn .Let a1,a2,…,an be the adjacent vertices of a0  and b1,b2,…,bn  be the adjacent vertices of b0.

EG=e1*,ei,eij,gi,gij here ei=b0bi,gi=a0ai,e1*=a0b0,eij=bibj,gij=aiaj .

Define ?:V?{0,1,2,…,Fq+1}

?a0=F2                                                              

?ai= F2i+1 +F2           for i= 1,2,3,…,n

?bi= F4n-2i+2                for i=1, 2,3,…,n

?b0=0

The edge labels are

ei=?*(b0bi)=?b0-?bi           for  i=1,2,3,…,n

gi=?*(a0ai)=?a0-?ai           for  i=1,2,3,…,n

e1*=?*(a0b0)=?a0-?b0          

eij=?*(bibj)=?bi-?bj           for  i=1,2,3,…,n   and for  j=i+1

gij=?*(aiaj)=?ai-?aj           for  i=1,2,3,…,n  and for  j=i+1

Here all the vertex  sums are pairwise distinct and all the edges are  unique.

Thus  ?  admitsFGAML .

Hence the graph BSn,n is FGAMG .

Python coding and output is enclosed in  the Appendix .

Illustration 3.4

Fig.4.Bi-Shell  BS4,4  is FGAMG

Conclusion

In this article the concept of Fibonacci Graceful Anti-Magic Labeling and demonstrated some Star related graphs are Fibonacci Graceful Anti – Magic Graphs. Python codings are generated for the Fibonacci Graceful Anti-Magic Labeling for the Star related Graphs. In future different concept of labeling will be developed.

References

[1] Ameenal Bibi.K and T.Ranjani “Fibonacci mean Anti-magic labeling of some graphs” Kong.Res.J. ISSN : 2349-2694 [2] Bondy J. A. and Murty U.S. R., Grap Theory with applications, Newyork Macmill Ltd. Press (1976). [3] David .W and Anthony .E. Barakaukas, “Fibonacci graceful graphs”. [4] D.Amuthavalli and M.Sangeetha “Super Fibonacci Gracefulness of shell related graphs”. International Journal of Mathematics Trends and Technology volume 53-No.6.January 2018. [5] D.Amuthavalli and M.Premkumar “Super Fibonacci Graceful labeling of cycle related graphs” . International Journal of Mathematics Trends and Technology volume 38-No.3.October 2016. [6] Joseph A. Gallian, “A Dynamic survey of graph labeling”, the electronic journal of combinatorics,(2022). [7] Sivaranjani.B and R.Kala “Fibonacci mean Anti-magic labeling of some graphs” Advances in Mathematics: Scientific Journal 9 (2020), no.5 2561-2572.ISSN: 1857-8365. [8] Solomon Golomb “Graph Theory and Computing”(Academic Press: 1972). [9] S.Somasundaram & T.Nicholas “on (a,d) -anti magic special trees, unicyclic graphs and complete bipartite graphs ars.com, 70,2004 207-220. [10] Tao-Ming wang and Cheng-Chih Hsiao “On anti-magic labeling for graphs products”, the ScienceDirect, Discrete Mathematics 308(2008) 3624-3633.

Copyright

Copyright © 2025 D. Amuthavalli , Dr. S. Murugesan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.