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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 G 1 () p i i EG . 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 2pG Keywords: energy of a graph, hyperenergetic, frucht graph, automorphism 1. Introduction The concept of hyperenergetic graphs was introduced by Gutman [1]. The existence of hyperenergetic graphs has been known for quite some time, their systematic design was first achieved by Walikar et al. [2]. Details can also be seen in [3–6]. The following result can be found in [4]. Theorem 1. A graph with p vertices and m edges such that is not hyperenergetic. 2mp2 In 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 {(,) : A ggh 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 (,) g h 1 gk is an arc, for some , that is kghh and 1 g k is called the color of (,) g k. 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 . 1 G(D ()CD The following is the construction of Frucht graph . 1 G Replace each arc ij g g of by a figure joining vertices D i g and j g . The figure consists of the 3-path iii j g uvg 21k , and two paths- a path (containing vertices) rooted at , and a path (containing 2k p p 2k i u21k vertices) rooted at , where i v1 ij k g g g is the color of ij g g. (Note that there will be a similar figure corresponding to j i g g for a different k). Clearly, the Frucht graph with 1()Gs has (1)(21)ns s vertices. Theorem 3. The Frucht graph has 1 G(2 1)ns s edges. Proof. The number of edges in is given by m1()G 11 4( 1) (41)(2 1) 2 ns ik ss mkn snss Theorem 4. The Frucht graph is not hyperener- getic. 1 G Proof. We observe that, (21)2(1)(2)222mnssns sp . Thus the result follows from Theorem 1. Lovasz [10] 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 i g and j g , in which to the first internal vertices are attached a path and to the last internal vertex (near k2 P j g ) is attached a path . 3 P Theorem 5. has vertices, where 2()G2 (41ns s) s . Proof. In each arc 2()Gij g g 4k is replaced by a figure with internal vertices(that is excluding 12kk i 3 3 g and j g ) if 1 ijk g gg . Therefore total number of extra vertices introduced to form is equal to 2()G 22 11 1 (23)(4 )(4 ) ns n ik i kssns s Hence 22 2 (())( 4)( 41)VGnssn nss . Theorem 6. has edges, where 2()G2 (5ns s) s . Proof. The number of edges in is given by 2()G 22 11 1 (22)( 5)( 5 ns n ik i mkk ssns )s 2 . 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 [1] I. Gutman, “The energy of a graph,” Journal of the Ser- bian 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,” Jour- nal 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 Ap- plications, 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. Copyright © 2009 SciRes IIM |