Analysis of Complex Networks

7.6 Concluding Remarks

At this moment the most significant open problem in the theory of graph energy seems to be the characterization of n-vertex graphs with the greatest energy. Although quite recently much progress has been achieved in this direction (cf. Theorem 7.5), the problem is still far from being completely solved. An additional difficulty that recently emerged [14] is the fact that for some values of n, there exist numerous maximum-energy n-vertex graphs.

There have been several recent attempts to extend the graph-energy concept to eigenvalues of matrices other than the adjacency matrix. Especially much work has been done on the so-called “Laplacian graph energy,” based on the spectrum of the Laplacian matrix, and on “distance graph energy,” based on the spectrum of the distance matrix. “Energy” has been redefined so that it could be associated with any matrix, including nonsquare matrices. However, the discussion of such energylike quantities goes beyond the scope of this survey.

References

1 Cvetković, D., Doob, M., Sachs, H., Spectra of Graphs – Theory and Application, Academic Press, New York, 1980.

2 Gutman, I., Trinajstić, N., Graph theory and molecular orbitals, Topics Curr. Chem. 42 (1973) 49–93.

3 Graovac, A., Gutman, I., Trinajstić, N., Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin, 1977.

4 Gutman, I., Polansky, O.E., Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.

5 Coulson, C. A., O'Leary, B., Mallion, ...