Geodesic
Hyperenergetic graphs and cyclomatic number
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 35 (2010), p. 1 .

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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. {\bf 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 $c \leq 8$\,.
Classification : 05C50
Keywords: energy (of graph), spectrum (of graph), hyperenergetic 
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     title = {Hyperenergetic graphs and  cyclomatic number},
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X. Shen; Y. Hou; I. Gutman; X. Hui. Hyperenergetic graphs and  cyclomatic number. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 35 (2010), p. 1 . http://geodesic.mathdoc.fr/item/BASS_2010_35_a0/