Frucht Graph is not Hyperenergetic

Author(s)
S. PIRZADA

ABSTRACT

If are the eigen values of a p-vertex graph , the energy of is . If , then is said to be hyperenergetic. We show that the Frucht graph, the graph used in the proof of well known Frucht’s theorem, is not hyperenergetic. Thus showing that every abstract group is isomorphic to the automorphism group of some non-hyperenergetic graph. AMS Mathematics Subject Classification: 05C50, 05C35

If are the eigen values of a p-vertex graph , the energy of is . If , then is said to be hyperenergetic. We show that the Frucht graph, the graph used in the proof of well known Frucht’s theorem, is not hyperenergetic. Thus showing that every abstract group is isomorphic to the automorphism group of some non-hyperenergetic graph. AMS Mathematics Subject Classification: 05C50, 05C35

Cite this paper

nullS. PIRZADA, "Frucht Graph is not Hyperenergetic,"*Intelligent Information Management*, Vol. 1 No. 2, 2009, pp. 120-121. doi: 10.4236/iim.2009.12017.

nullS. PIRZADA, "Frucht Graph is not Hyperenergetic,"

References

[1] I. Gutman, “The energy of a graph,” Journal of the Serbian Chemical Society, Vol. 64, pp. 199–205, 1999.

[2] L. Lovasz, “Combinatorial problems and exercises,” North-Holland, Amsterdam, 1979.

[3] I. Gutman and L. Pavlovic, “The energy of some graphs with large number of edges,” Bull. Acad. Serbe Sci. Arts, Vol. 118, pp. 35–50, 1999.

[4] I. Gutman Y. Hou, H. B. Walikar, H. S. Ramane, and P. R. Hamphiholi, “No huckel graph is hyperenergetic,” Journal of the Serbian Chemical Society, Vol. 65, No. 11, pp. 799–801, 2000.

[5] I. Gutman, “The energy of a graph, old and new results,” In: A. Betten, et al., Algebraic Combinatorics and its Applications, Springer-Verlag, Berlin, pp.196–211, 2001.

[6] I. Gutman and L. Pavlovic, “The energy of some graphs with large number of edges,” Bull. Acad. Serbe Sci. Arts, Vol. 118, pp. 35–50, 1999.

[7] R. Frucht, “Herstellung von graphin mit vorgege bener abstakten Gruppe,” Compositio Mathematica, Vol. 6, pp. 239–250, 1938.

[8] R. Frucht, “Graphs of degree three with a given abstract group,” Canadian Journal Mathematics, Vol. 1, pp. 365–378, 1949.

[9] R. Frucht and F. Harary, “On the corona of two graphs,” Aequationes mathematicae, Basel, Vol. 4, pp. 322–325, 1970.

[10] J. Koolen, V. Moulton, I. Gutman, and D. Vidovic, “More hyperenergetic molecular graphs,” Journal of the Serbian Chemical Society, Vol. 65, pp. 571–575, 2000.

[11] H. B. Walikar, H. S. Ramane, and P. R. Hamphiholi, “On the energy of a graph,” Proceedings of Conference on Graph Connections (R. Balakrishnan et al. eds.), Allied Publishers, New Delhi, pp. 120–123, 1999.

[1] I. Gutman, “The energy of a graph,” Journal of the Serbian Chemical Society, Vol. 64, pp. 199–205, 1999.

[2] L. Lovasz, “Combinatorial problems and exercises,” North-Holland, Amsterdam, 1979.

[3] I. Gutman and L. Pavlovic, “The energy of some graphs with large number of edges,” Bull. Acad. Serbe Sci. Arts, Vol. 118, pp. 35–50, 1999.

[4] I. Gutman Y. Hou, H. B. Walikar, H. S. Ramane, and P. R. Hamphiholi, “No huckel graph is hyperenergetic,” Journal of the Serbian Chemical Society, Vol. 65, No. 11, pp. 799–801, 2000.

[5] I. Gutman, “The energy of a graph, old and new results,” In: A. Betten, et al., Algebraic Combinatorics and its Applications, Springer-Verlag, Berlin, pp.196–211, 2001.

[6] I. Gutman and L. Pavlovic, “The energy of some graphs with large number of edges,” Bull. Acad. Serbe Sci. Arts, Vol. 118, pp. 35–50, 1999.

[7] R. Frucht, “Herstellung von graphin mit vorgege bener abstakten Gruppe,” Compositio Mathematica, Vol. 6, pp. 239–250, 1938.

[8] R. Frucht, “Graphs of degree three with a given abstract group,” Canadian Journal Mathematics, Vol. 1, pp. 365–378, 1949.

[9] R. Frucht and F. Harary, “On the corona of two graphs,” Aequationes mathematicae, Basel, Vol. 4, pp. 322–325, 1970.

[10] J. Koolen, V. Moulton, I. Gutman, and D. Vidovic, “More hyperenergetic molecular graphs,” Journal of the Serbian Chemical Society, Vol. 65, pp. 571–575, 2000.

[11] H. B. Walikar, H. S. Ramane, and P. R. Hamphiholi, “On the energy of a graph,” Proceedings of Conference on Graph Connections (R. Balakrishnan et al. eds.), Allied Publishers, New Delhi, pp. 120–123, 1999.