EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques
Vol. CXLI, No. 35, pp. 1–8 (2010)

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Hyperenergetic graphs and cyclomatic number

X. Shen, Y. Hou, I. Gutman and X. Hui

College of Mathematics and Computer Science, Hunan Normal University, Hunan 410081, P. R. China
Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia

Abstract: Let $G$ be a graph with $n$ vertices and $m$ edges. Then its cyclomatic number is $c=m-n+1$ . If $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $G$ , then its energy is $E(G)=\sum_{i=1}^n |\lambda_i|$ . The graph $G$ is said to be hyperenergetic if $E(G)>E(K_n)=2n-2$ . It is known [Nikiforov, J. Math. Anal. Appl. 327 (2007) 735–738] that almost all graphs are hyperenergetic. We now show that for any $c<\infty$ , there is only a finite number of hyperenergetic graphs with cyclomatic number $c$ . In particular, there are no hyperenergetic graphs with $c \leq 8$ .

Keywords: energy (of graph), spectrum (of graph), hyperenergetic

Classification (MSC2000): 05C50

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Electronic fulltext finalized on: 3 Oct 2010. This page was last modified: 20 Jun 2011.

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