Closed and asymptotic formulas for energy of some circulant graphs

# Closed and asymptotic formulas for energy of some circulant graphs

David Blázquez-Sanz  and  Carlos Alberto Marín Arango Universidad Nacional de Colombia - Sede Medellín Facultad de Ciencias Escuela de Matemáticas Medellín, Colombia Instituto de Matemáticas Universidad de Antioquia Medellín, Colombia
###### Abstract.

We consider circulant graphs where the vertices are the integers modulo and the neighbours of are . The energy of is a trigonometric sum of terms. For low values of we compute this sum explicitly. We also study the asymptotics of the energy of 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 terms. As an application we show that in the family 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 . Finally we show that the graph minimizes the energy among all the regular graphs of degree .

Keywords: Circulant graph, Graph energy, Finite Fourier transform. 2010 MSC 05C35; 05C50; 42A05.

## 1. Introduction

Let be a graph with vertices and eigenvalues . The energy of is defined as the sum of the absolute values of its eigenvalues:

 E(G)=N∑j=1|λj|.

This concept was introduced in the mathematical literature by Gutman in 1978, [2]. For further details on this theory we refer to [6] and the literature cited in.

The -vertex complete graph has eigenvalues y ( times). Therefore, . In [2], it was conjectured that the complete graph has the largest energy among all -vertex graphs , that is, with equality iff . By means of counterexamples, this conjecture was shown to be false, see for instance [11].

An -vertex graph whose energy satisfies is called hyperenergetic. These graphs has been introduced in [3] in relation with some problems of molecular chemistry. A simplest construction of a family of hyperenergetic graphs is due to Walikar et al, [10], where authors showed that the line graph of , is hyperenergetic. There are a number of other recent results on hyperenergetic graphs [1, 5]. There are also some recent results for the hyperenergetic circulant graphs, see for instance [7, 9].

In this paper we consider the following family of circulant graphs. For each and with we define the circulant graph with vertices in which there is an edge between and if the inequation has a solution , see fig. 1. We give a closed formula for the energy of the circulant graph . Based on that formula we also show that the energy of the graph obtaining by deleting the edges of a perfect matching of a complete graph of index is non-hyperenegetic. Moreover, we show that the graph has minimal energy over the set of regular graphs of degree . Finally, we find some bounds and explicit expressions for . This allows us to conclude that for there is finite number of hyperenergetic graphs of the form ; moreover, for there is a finite number of non-hyperenergetic graphs of the form .

## 2. Preliminaries

Let us remark the following easy to check facts:

• is the -cycle graph.

• is the complete graph .

• is the complement of a perfect matching of the edges in a complete graph .

• is the complement of a hamiltonian cycle in a complete graph .

The eigenvalues of can be computed by terms of a finite Fourier transform, it turns out that if denotes the -th eigenvalue of then;

 λ(r,N,k)=u(r,2kπN)

where

 u(r,θ)=2r∑m=1cos(mθ).

Thus, the energy for the circulant graph is given by the expression:

 (1) E(r,N)=2N−1∑k=0∣∣ ∣∣r∑m=1cos(2kmπN)∣∣ ∣∣.

## 3. Closed formula for energy

For small values of , we can split the trigonometric sum (1) defining into positive and negative parts:

For each we have a finite number of trigonometric sums along arithmetic sequences. Those sums can be computed explicitly, by means of the following elementary Lemma on trigonometric sums.

###### Lemma 1.

Let be an arithmetic sequence of real numbers with . The following identities hold:

 n∑k=0cos(ak)=sin(an+r/2)−sin(a0−r/2)2sin(r/2),
 n∑k=0sin(ak)=cos(a0−r/2)−cos(an+r/2)2sin(r/2).
###### Proof.

Let us take . We have and . Each sum split as the addition of two geometric sums that are explicitly computed. By undoing the same change of variables, we obtain the above identities. ∎

For we obtain a closed formula for the energy of the cycle:

 (2) E(1,N)=4sin(πN(2⌊N4⌋+1))sin(πN)

In the case , we take into account:

 |cos(x)+cos(2x)|={cos(x)+cos(2x)si −π3≤x≤π3−cos(x)−cos(2x)si π3≤x≤5π3

and thus we obtain:

 (3) E(2,N)=4⎡⎢ ⎢⎣sin(πN(2⌊N6⌋+1))sin(πN)+sin(2πN(2⌊N6⌋+1))sin(2πN)⎤⎥ ⎥⎦.

Figure 2 illustrates the modular behavior (mod and mod respectively) of the energy which is easily seen in formulae (2) and (3). Analogous formula, of growing complexity, may be computed for other values of .

By means of Lemma 1 we can also compute explicitly the energy of the graphs obtaining:

 E(r,2r+2)=2r+2r+1∑k=1∣∣ ∣ ∣∣sin(kπ−kπ2r+2)sin(kπ2r+2)−1∣∣ ∣ ∣∣=
 =2r+2r+1∑k=1∣∣(−1)k+1−1∣∣=4r

Let us recall that the graph is isomorphic to ; the graph obtained by deleting a perfect matching from a complete graph of even order . It is a regular graph of degree . It is well know that the energy of a regular graph of degree is equal or greater than (see [4, pag. 77]). This trivial lower bound is reached for the complete graph . The family gives us another example of regular graphs minimizing energy within each family of regular graphs of fixed even degree.

###### Theorem 1.

Let us consider the the graph obtained by deleting the edges of a perfect matching of a complete graph of index .

• and thus is non-hyperenergetic.

• minimizes the energy in the family of regular graphs of degree .

## 4. Asymptotic behaviour

A direct consequence of Theorem 2 in [7] is that the limit exists and it is bigger than . Here we compute it explicitly.

###### Theorem 2.

For each the limit:

 limN→∞E(r,N)N−1=Ir

has value:

 Ir=1π∫π0∣∣ ∣ ∣∣sin((12+r)θ)sin(θ2)−1∣∣ ∣ ∣∣dθ.

Note that the integral is the -norm of where is the Dirichlet kernel. From the triangular inequality we obtain:

 Lr−1≤Ir≤Lr+1

where is the Lebesgue constant (see [12] page 67).

We may evaluate analytically integral. Equation

 u(r,θ)=0

can be solved analytically. It yields

 θ=2mπr,θ=(2m+1)πr+1,m∈Z.

Between and we obtain the partition:

 0,πr+1,2πr,3πr+1,4πr,…,π.

The function is positive in the first interval, negative in the second interval, and so on. This allows us to evaluate the integral obtaining,

 Ir=4πr∑k=1⌊r2⌋∑m=0sin((2m+1)kπr+1)−sin(2mkπr)k

and finally by application of Lemma 1:

 Ir=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩2πr∑k=11k⎡⎢ ⎢⎣1−cos(rkπr+1)sin(kπr+1)+cos(kπr)−(−1)ksin(kπr)⎤⎥ ⎥⎦ if r odd,2πr∑k=11k⎡⎢ ⎢⎣1−(−1)ksin(kπr+1)+cos(kπr)−cos((r+1)kπr)sin(kπr)⎤⎥ ⎥⎦ if r even.

The first values can be computed explicitly obtaining,

 I1=4π,I2=3√3π,I3=16√2−3√33π

and some approximate values are:

 I4≃1.985…,I5≃2.087…,I6≃2.170…

In particular we have that for and for . It follows the following result.

###### Theorem 3.

The following statements hold:

• For there is a finite number of hyperenergetic graphs of the form .

• For each there is a finite number on non-hyperenergetic graphs of the form . An example is given by .

A numerical exploration (see fig. 3) allows us to find all the hyperenergetic graphs of the form with .

1. There is no hyperenergetic graph of the form or .

2. There are five hyperenergectic graphs of the form with .

3. There is much bigger but finite family of hyperenergetic graphs of the form , ilustrated by fig. 3 (right).

With respect to Theorem 3 (b), we conjecture that the only non-hyperenergetic graph of the form with is the graph .

## Acknowledgements

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

## References

• [1] R. Balakrishnan, The energy of a graph, Linear Algebra and its Applications 387 (2004) 287–295.
• [2] I. Gutman, The energy of a graph. Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978) 1-22.
• [3] I. Gutman, Hyperenergetic molecular graphs, J. Serb. Chem. Soc. 64 (1999) 199–205.
• [4] I. Gutman, Hyperenergetic and hypoenergetic Graphs, Zbornik Radova, 4(22) 2011, Pages 113–135.
• [5] J.H. Koolen, V. Moulton, I. Gutman, D. Vidović, More hyperenergetic molecular graphs.J. Serb. Chem. Soc. 65, 571–575 (2000)
• [6] X. Li, Y. Shi, I. Gutman, Graph energy, Springer, New York, 2012.
• [7] I. Shparlinski, On the energy of some circulant graphs Linear Algebra and its Applications, Volume 414, Issue 1, 1 April 2006, Pages 378–382.
• [8] J.D. Stegeman, On the constant in the Littlewood problem, Math. Ann., 261 (1982) Pages 51–54.
• [9] D. Stevanović, I. Stanković, Remarks on hyperenergetic circulant graphs, Linear Algebra and its Applications Volume 400, 1 May 2005, Pages 345–348.
• [10] H.B. Walikar, H.S. Ramane, P.R. Hampiholi, in On the Energy of a Graph, ed. by R. Balakrishnan, H.M. Mulder, A. Vijayakumar. Graph Connections (Allied, New Delhi, 1999), 120–-123
• [11] H.B. Walikar, I. Gutman, P.R. Hampiholi, H.S. Ramane, Non-hyperenergetic graphs, Graph Theory Notes New York 41 (2001) 14–-16.
• [12] A. Zygmund, Trigonometric Series, Vols. I & II Combined, Cambridge University Press, 3rd Edition, 2002.
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