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

Hyperenergetic graphs and cyclomatic numberX. Shen, Y. Hou, I. Gutman and X. HuiCollege of Mathematics and Computer Science, Hunan Normal University, Hunan 410081, P. R. ChinaFaculty 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 $c=mn+1$ . 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 $E(G)>E(K_n)=2n2$ . 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 Full text of the article: (for faster download, first choose a mirror)
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