 Intelligent Information Management, 2009, 1, 120-121 doi:10.4236/iim.2009.12017 Published Online November 2009 (http://www.scirp.org/journal/iim) Copyright © 2009 SciRes IIM Frucht Graph is not Hyperenergetic S. PIRZADA Department of Mathematics, University of Kashmir, Kashmir, India Email: sdpirzada@yahoo.co.in Abstract: If 12, ,...,p are the eigen values of a p-vertex graph , the energy of is G G1()piiEG. 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 Clas-sification: 05C50, 05C35 () 2EG 2pG Keywords: energy of a graph, hyperenergetic, frucht graph, automorphism 1. Introduction The concept of hyperenergetic graphs was introduced by Gutman . The existence of hyperenergetic graphs has been known for quite some time, their systematic design was first achieved by Walikar et al. . Details can also be seen in [3–6]. The following result can be found in . Theorem 1. A graph with p vertices and m edges such that is not hyperenergetic. 2mp2In this paper, we give the existence of one more class of hyperenergetic graphs, called the Frucht graphs. 2. Frucht Graph is not Hyperenergetic Let Γ be a group with n elements and Λ be a set of Γ not containing the identity e. The Cayley digraph is defined to be the digraph with vertex set VГ and arc set {(,) :Aggh hKKΛ}. It is denoted by Γ, Λ). If Λ = Γ -e, the resulting Cayley digraph is complete and is denoted by . If Λ is a set of generators for Γ, the Cayley digraph is called the basic Cayley digraph. (DD(, )In the Cayley digraph (DDΓ, Λ), if (,)gh1gk is an arc, for some , that is kghh and 1gk is called the color of (,)gk. An automorphism of is said to be color preserv-ing if it preserves the colors of the arcs. It is well known that the group of color preserving automor-phisms of the Cayley digraph Γ, Λ) is isomor-phic to Γ. D()CD(DDThe following result is the Frucht’s Theorem [7–9]. Theorem 2. Every group is isomorphic to the auto-morphism group of some graph. While proving Theorem 2, Frucht obtained the graph from Γ, Λ) called as Frucht graph, whose auto-morphism group is isomorphic to . 1G(D()CDThe following is the construction of Frucht graph . 1GReplace each arc ijgg of by a figure joining vertices Dig and jg. The figure consists of the 3-path iii jguvg21k, and two paths- a path (containing vertices) rooted at , and a path (containing 2kpp2kiu21k vertices) rooted at , where iv1ij kggg is the color of ijgg. (Note that there will be a similar figure corresponding to jigg for a different k). Clearly, the Frucht graph with 1()Gs has (1)(21)ns s vertices. Theorem 3. The Frucht graph has 1G(2 1)ns s edges. Proof. The number of edges in is given by m1()G114( 1)(41)(2 1)2nsikssmkn snss  Theorem 4. The Frucht graph is not hyperener-getic. 1GProof. We observe that, (21)2(1)(2)222mnssns sp . Thus the result follows from Theorem 1. Lovasz  gives an alternate construction of the J. WU ET AL. 121 graph used to prove the Fruchts theorem. In this case the figure of color is a path of length 2()Gk2k, including the end vertices ig and jg, in which to the first internal vertices are attached a path and to the last internal vertex (near k2Pjg) is attached a path . 3PTheorem 5. has vertices, where 2()G2(41ns s)s . Proof. In each arc 2()Gijgg4k is replaced by a figure with internal vertices(that is excluding 12kki3 3 g and jg) if 1ijkggg. Therefore total number of extra vertices introduced to form is equal to 2()G2211 1(23)(4 )(4 )ns nik ikssns   s Hence 222(())( 4)( 41)VGnssn nss . Theorem 6. has edges, where 2()G2(5ns s)s . Proof. The number of edges in is given by 2()G2211 1(22)( 5)( 5ns nik imkk ssns  )s2. Theorem 7. is not hyperenergetic. 2()GProof. We see that 22(5)2(41)22mns snssp . Thus the result follows from Theorem 1. Combining the above observations, we conclude with the following result. Theorem 8. Every group is isomorphic to the auto-morphism group of some non-hyperenergetic graph. REFERENCES  I. Gutman, “The energy of a graph,” Journal of the Ser-bian Chemical Society, Vol. 64, pp. 199–205, 1999.  L. Lovasz, “Combinatorial problems and exercises,” North-Holland, Amsterdam, 1979.  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.  I. Gutman Y. Hou, H. B. Walikar, H. S. Ramane, and P. R. Hamphiholi, “No huckel graph is hyperenergetic,” Jour-nal of the Serbian Chemical Society, Vol. 65, No. 11, pp. 799–801, 2000.  I. 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Balakrishnan et al. eds.), Allied Publishers, New Delhi, pp. 120–123, 1999. Copyright © 2009 SciRes IIM