Hyperenergetic graphs and cyclomatic number

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

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 $c \leq 8$\,.