Closed and asymptotic formulas for energy of some circulant graphs

Abstract

We consider circulant graphs G(r, N) where the vertices are the integers modulo N and the neighbours of 0 are . The energy of G(r, N) is a trigonometric sum of terms. For low values of r we compute this sum explicitly. We also study the asymptotics of the energy of G(r, N) for . There is a known integral formula for the linear growth coefficient, we find a new expression of the form of a finite trigonometric sum with r terms. As an application we show that in the family G(r, N) for there is a finite number of hyperenergetic graphs. On the other hand, for each there is at most a finite number of non-hyperenergetic graphs of the form G(r, N). Finally, we show that the graph minimizes the energy among all the regular graphs of degree 2r.

Keywords:

• Circulant graph
• graph energy
• finite fourier transform

AMS Subject Classifications:

• 05C35
• 05C50
• 42A05

Acknowledgements

We want to thank Juan Pablo Rada for introducing us to the topic of graph energy, and sharing his experience with us. We also thank the anonymous referees for helping to improve the presentation and quality of the paper. The first author acknowledges Universidad Nacional de Colombia research grant HERMES-27984. The second author acknowledges the support of Universidad de Antioquia.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Universidad Nacional de Colombia research [grant HERMES-27984]; Universidad de Antioquia.