A formula for the energy of circulant graphs with two generators

In this note, we derive closed formulas for the energy of circulant graphs generated by $1$ and $\gamma$, where $\gamma\geqslant2$ is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.

Let 1 γ 1 · · · γ d be integers. The circulant graph C γ1,...,γ d n generated by γ 1 , . . . , γ d on n vertices labelled 0, 1, . . . , n − 1, is the 2d-regular graph such that for all v ∈ Z/nZ, v is connected to v + γ i mod n and to v − γ i mod n, for all i = 1, . . . , d. The adjacency matrix A = (A ij ) of a graph on n vertices is the n × n matrix with rows and columns indexed by the vertices such that A ij is the number of edges connecting vertices i and j. Let λ k , k = 1, . . . , n, denote the eigenvalues of the adjacency matrix. The energy of a graph G on n vertices is defined by the sum of the absolute value of the eigenvalues of A, that is The energy of circulant graphs and integral circulant graphs is widely studied, see for example [4,5,6,7]. It has interesting applications in theoretical chemistry, namely, it is related to the π-electron energy of a conjugated carbon molecule, see [2]. In the following theorem, we give a formula for the energy of circulant graphs with two generators, 1 and γ, γ 2. The formula is interesting as n is larger than γ.
Theorem. Let D n (x) denote the Dirichlet kernel. The energy of the circulant graph C 1,2 n is given by E(C 1,2 n ) = 4 D n/6 (2π/n) + D n/6 (4π/n) . For γ 3, the energy of the circulant graph C 1,γ n is given by where x denotes the greatest integer smaller or equal to x and x denotes the smallest integer greater or equal to x.
The sum of cos(kx) over consecutive k's can be expressed in terms of the Dirichlet kernel, namely As a consequence, The energy of C 1,2 n is thus given by The formula then follows from the fact that for odd n, D (n−1)/2 (2πm/n) = 0 for m = 1, 2, and for even n, D n/2−1 (2π/n) = 1 and D n/2−1 (4π/n) = −1.
Writing the above relation in terms of Dirichlet kernels, it comes Hence The formula follows from the fact that D n/2 (2πm/n) = 0 for m = 1, γ. Let n be even. As for the case when n is odd, we write the energy as follow (cos(2πk/n) + cos(2πγk/n)).
A graph is called hyperenergetic if his energy is greater than the one of the complete graph K n . The eigenvalues of K n are given by n − 1 and −1 with multiplicity n − 1, so that his energy is given by E(K n ) = 2(n − 1). The figure on the left below shows how the energy of C 1,γ n grows with respect to n for γ = 8. We see that it is not hyperenergetic and that the energy grows more or less linearly with respect to n. The figure on the right shows the energy of C 1,γ n with fixed n as γ varies. We observe that the energy stays more or less constant independently of γ. As a consequence of the theorem, we can carry out the sum of the Dirichlet kernels when the number of vertices is proportional to 2(γ − 1)(γ + 1).
In [1], the author considered the graphs K n −H where K n is the complete graph on n vertices and H is a Hamilton cycle of K n and asked whether these graphs are hyperenergetic. In [7], the author showed that the energy of K n − H is given by |1 + 2 cos(2πk/n)| and that as n goes to infintiy, it is hyperenergetic. In the following proposition we give a formula for it for all n 3.
By elementary analysis, one can show that E(K n − H) − 2(n − 1) is increasing in n. As a consequence, we find that K n − H are hyperenergetics for all n 10. This has been previously found in [7].