Closed and asymptotic formulas for energy of some circulant graphs

TL;DR

This paper derives explicit and asymptotic formulas for the energy of circulant graphs G(r,N), revealing properties about hyperenergetic graphs and energy minimization among regular graphs.

Contribution

It provides new explicit formulas and asymptotic expressions for the energy of G(r,N), and characterizes hyperenergetic and energy-minimizing graphs within this family.

Findings

Explicit formulas for energy when r is small.

Asymptotic formulas for large N.

Characterization of hyperenergetic graphs for r ≤ 4.

Abstract

We consider circulant graphs G(r,N) where the vertices are the integers modulo N and the neighbours of 0 are {-r,...,-1,1,...,r}. The energy of G(r,N) is a trigonometric sum of N*r terms. For low values of r we compute this sum explicitly. We also study the asymptotics of the energy of G(r,N) for big N. 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 r less or equal than 4 there is a finite number of hyperenergetic graphs. On the other hand, for each r>4 there is at most a finite number of non-hyperenergetic graphs of the form G(r,N). Finally we show that the graph G(r,2r+1) minimizes the energy among all the regular graphs of degree 2r.