Abstract
We derive closed formulas for the energy of circulant graphs generated by and , where is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.
Let be integers. The circulant graph generated by on vertices labelled is the 2D-regular graph such that, for all , is connected to and to , for all . The adjacency matrix of a graph on vertices is the matrix with rows and columns indexed by the vertices such that is the number of edges connecting vertices and . Let , , denote the eigenvalues of the adjacency matrix. The energy of a graph on vertices is defined by the sum of the absolute values of the eigenvalues of ; that is, The energy of circulant graphs and integral circulant graphs is widely studied; see, for example, [1–4]. It has interesting applications in theoretical chemistry; namely, it is related to the -electron energy of a conjugated carbon molecule; see [5]. In the following theorem, we give a formula for the energy of circulant graphs with two generators, and , . The formula is interesting as is larger than .
Theorem 1. Let denote the Dirichlet kernel. The energy of the circulant graph is given by For , the energy of the circulant graph is given by where denotes the greatest integer smaller than or equal to and denotes the smallest integer greater than or equal to .
Proof. The adjacency matrix of a circulant graph is circulant; it follows that the eigenvalues of are given by , (see [6]). The energy of is then given by Let . The two roots of the equation for are and . We write the energy asThe sum of over consecutive ’s can be expressed in terms of the Dirichlet kernel; namely, As a consequence, The energy of is thus given by The formula then follows from the fact that, for odd , for , and, for even , and .
Let . For odd , the solutions of the equation for are given in the increasing order by For even , they are given by . Let be odd. We split the sum over of cosines to group the positive terms together and the negative terms together. The energy is given byWriting the above relation in terms of Dirichlet kernels, we haveHence,The formula follows from the fact that for .
Let be even. As for the case when is odd, we write the energy as follows: where if is odd and otherwise.
For even , relations (9), (10), and (11) also hold. The theorem then follows from the fact that and . For odd , we have Expressing it in terms of Dirichlet kernels, we have The theorem follows from the fact that for .
A graph is called hyperenergetic if its energy is greater than the one of the complete graph . The eigenvalues of the adjacency matrix of are given by and with multiplicity , so that its energy is given by .
Figure 1(a) shows how the energy of grows with respect to for . We see that it is not hyperenergetic and that the energy grows more or less linearly with respect to . Figure 1(b) shows the energy of with fixed as varies. We observe that the energy stays more or less constant independently of .
(a) Energy of (crosses) and of (circles) with respect to
(b) Energy of with respect to
As a consequence of the theorem, we can carry out the sum of the Dirichlet kernels when the number of vertices is proportional to .
Corollary 2. Given integers and , the energy of the circulant graph is given by
Proof. Let and be integers. The sum over of Dirichlet kernels of index is given by By multiplying the summation by and using the trigonometric identity , we have The corollary then follows by applying the above relation first with , and second with , , and , .
In [7], the author considered the graphs , where is the complete graph on vertices and is a Hamilton cycle of , and asked whether these graphs are hyperenergetic. In [4], the authors showed that the energy of is given by and that as goes to infinity, it is hyperenergetic. In the following proposition, we give a formula for it for all .
Proposition 3. For all , the energy of is given by
Proof. We have Since , the proposition follows.
By elementary analysis, one can show that is increasing in . As a consequence, we find that are hyperenergetic for all . This has been previously found in [4].
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
The author acknowledges support from the Swiss NSF Grant no. 200021_132528/1.