Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The Line n-sigraph of a Symmetric n-sigraph-V

  • Reddy, P. Siva Kota (Department of Mathematics, Siddaganga Institute of Technology) ;
  • Nagaraja, K.M. (Department of Mathematics, JSS Academy of Technical Education) ;
  • Geetha, M.C. (Department of Mathematics, East West Institute of Technology)
  • Received : 2012.02.21
  • Accepted : 2013.05.08
  • Published : 2014.03.23
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Abstract

An n-tuple ($a_1,a_2,{\ldots},a_n$) is symmetric, if $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$. Let $H_n$ = {$(a_1,a_2,{\ldots},a_n)$ ; $a_k$ ${\in}$ {+,-}, $a_k$ = $a_{n-k+1}$, $1{\leq}k{\leq}n$} be the set of all symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair $S_n$ = (G,${\sigma}$) ($S_n$ = (G,${\mu}$)), where G = (V,E) is a graph called the underlying graph of $S_n$ and ${\sigma}$:E ${\rightarrow}H_n({\mu}:V{\rightarrow}H_n)$ is a function. The restricted super line graph of index r of a graph G, denoted by $\mathcal{R}\mathcal{L}_r$(G). The vertices of $\mathcal{R}\mathcal{L}_r$(G) are the r-subsets of E(G) and two vertices P = ${p_1,p_2,{\ldots},p_r}$ and Q = ${q_1,q_2,{\ldots},q_r}$ are adjacent if there exists exactly one pair of edges, say $p_i$ and $q_j$, where $1{\leq}i$, $j{\leq}r$, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph $S_n$ = (G,${\sigma}$) as a symmetric n-sigraph $\mathcal{R}\mathcal{L}_r$($S_n$) = ($\mathcal{R}\mathcal{L}_r(G)$, ${\sigma}$'), where $\mathcal{R}\mathcal{L}_r(G)$ is the underlying graph of $\mathcal{R}\mathcal{L}_r(S_n)$, where for any edge PQ in $\mathcal{R}\mathcal{L}_r(S_n)$, ${\sigma}^{\prime}(PQ)$=${\sigma}(P){\sigma}(Q)$. It is shown that for any symmetric n-sigraph $S_n$, its $\mathcal{R}\mathcal{L}_r(S_n)$ is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs $S_n$ for which $\mathcal{R}\mathcal{L}_r(S_n)$~$\mathcal{L}_r(S_n)$ and $$\mathcal{R}\mathcal{L}_r(S_n){\sim_=}\mathcal{L}_r(S_n)$$, where ~ and $$\sim_=$$ denotes switching equivalence and isomorphism and $\mathcal{R}\mathcal{L}_r(S_n)$ and $\mathcal{L}_r(S_n)$ are denotes the restricted super line symmetric n-sigraph of index r and super line symmetric n-sigraph of index r of $S_n$ respectively.

Keywords

References

  1. K. S. Bagga, L. W. Beineke and B. N. Varma, Super line graphs, In: Y. Alavi, A. Schwenk (Eds.), Graph Theory, Combinatorics and Applications, vol. 1, Wiley-Interscience, New York, 1995, pp. 35-46.
  2. F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.
  3. V. Lokesha, P. S. K. Reddy and S. Vijay, The triangular line n-sigraph of a symmetric n-sigraph, Advn. Stud. Contemp. Math., 19(1)(2009), 123-129.
  4. K. Manjula, Some results on generalized line graphs, Ph.D. thesis, Bangalore University, Bangalore, 2004.
  5. E. Prisner, Graph Dynamics, Longman, London, 1995.
  6. R. Rangarajan and P. Siva Kota Reddy, Notions of balance in symmetric n-sigraphs, Proceedings of the Jangjeon Math. Soc., 11(2)(2008), 145-151.
  7. R. Rangarajan, P. S. K. Reddy and M. S. Subramanya, Switching Equivalence in Symmetric n-Sigraphs, Adv. Stud. Comtemp. Math., 18(1)(2009), 79-85.
  8. R. Rangarajan, P. S. K.Reddy and N. D. Soner, Switching equivalence in symmetric n-sigraphs-II, J. Orissa Math. Sco., 28(1 & 2)(2009), 1-12.
  9. P. S. K. Reddy and B. Prashanth, Switching equivalence in symmetric n-sigraphs-I, Advances and Applications in Discrete Mathematics, 4(1)(2009), 25-32.
  10. P. S. K. Reddy, S. Vijay and B. Prashanth, The edge $C_4$ n-sigraph of a symmetric n-sigraph, Int. Journal of Math. Sci. & Engg. Appls., 3(2)(2009), 21-27.
  11. P. S. K. Reddy, V. Lokesha and Gurunath Rao Vaidya, The Line n-sigraph of a symmetric n-sigraph-II, Proceedings of the Jangjeon Math. Soc., 13(3)(2010), 305-312.
  12. P. S. K. Reddy, V. Lokesha and Gurunath Rao Vaidya, The Line n-sigraph of a symmetric n-sigraph-III, Int. J. Open Problems in Computer Science and Mathematics, 3(5)(2010), 172-178.
  13. P. S. K. Reddy, V. Lokesha and Gurunath Rao Vaidya, Switching equivalence in symmetric n-sigraphs-III, Int. Journal of Math. Sci. & Engg. Appls., 5(1)(2011), 95-101.
  14. P. S. K. Reddy, M. C. Geetha and K. R. Rajanna, Switching equivalence in symmetric n-sigraphs-IV, Scientia Magna, 7(3)(2011), 34-38.
  15. P. S. K. Reddy, M. C. Geetha and K. R. Rajanna, Switching equivalence in symmetric n-sigraphs-V, International J. Math. Combin., 3(2012), 58-63.
  16. P. S. K. Reddy, K. M. Nagaraja and M. C. Geetha, The Line n-sigraph of a symmetric n-sigraph-IV, International J. Math. Combin., 1(2012), 106-112.
  17. E. Sampathkumar, P. S. K. Reddy, and M. S. Subramanya, Jump symmetric n-sigraph, Proceedings of the Jangjeon Math. Soc., 11(1)(2008), 89-95.
  18. E. Sampathkumar, P. S. K. Reddy, and M. S. Subramanya, The Line n-sigraph of a symmetric n-sigraph, Southeast Asian Bull. Math., 34(5)(2010), 953-958.