Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Graph Equations Involving Tensor Product of Graphs

  • Patil, H.P. (Department of Mathematics, Pondicherry University, Kalapet Campus) ;
  • Raja, V. (Department of Mathematics, Pondicherry University, Kalapet Campus)
  • Received : 2015.02.24
  • Accepted : 2017.01.21
  • Published : 2017.06.23
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Abstract

In this paper, we solve the following four graph equations $L^k(G)=H{\oplus}J$; $M(G)=H{\oplus}J$; ${\bar{L^k(G)}}=H{\oplus}J$ and ${\bar{M(G)}}=H{\oplus}J$, where J is $nK_2$ for $n{\geq}1$. Here, the equality symbol = means the isomorphism between the corresponding graphs. In particular, we shall obtain all pairs of graphs (G, H), which satisfy the above mentioned equations, upto isomorphism.

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References

  1. M. Behzad and G. Chartrand, Introduction to the theory of graphs, Allyn and Bacon. Inc. Boston, 1972.
  2. D. M. Cvetkovic, Spectrum of the total graph of a graph, Publ. Inst. Math. (Beograd), 16(30)(1973), 49-52.
  3. M. Farzan and D. A. Waller, Kronecker products and local joins of graphs, Canad. J. Math., 29(2)(1977), 255-269. https://doi.org/10.4153/CJM-1977-027-1
  4. T. Hamada and I. Yoshimura, Traversability and connectivity of the middle graph of a graph, Discrete Math., 14(1976), 247-255. https://doi.org/10.1016/0012-365X(76)90037-6
  5. F. Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass-Menlo Park, Calif.-London, 1969.
  6. V. R. Kulli and H. P. Patil, Graph equations for line graphs, middle graphs and entire graphs, J. Karnatak Univ. Sci., 23(1978), 25-28.
  7. H. P. Patil and R. Pandiya Raj, Characterizations of k-ctrees and graph valued functions, J. Combin. Math. Combin. Comput., 84(2013), 91-98.
  8. H. P. Patil and V. Raja, On tensor product of graphs, girth and triangles, Iran. J. Math. Sci. Inform., 10(1)(2015), 139-147.
  9. E. Sampathkumar and S. B. Chikkodimath, Semi-total graphs of a graph I, J. Karnatak Univ. Sci., 18(1973), 274-280.
  10. D. V. S. S. Sastry and B. S. P. Raju, Graph equations for line graphs, middle graphs and quasitotal graphs, Discrete Math., 48(1984), 113-119. https://doi.org/10.1016/0012-365X(84)90137-7
  11. P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc., 13(1962), 47-52. https://doi.org/10.1090/S0002-9939-1962-0133816-6