# The exact maximal energy of integral circulant graphs with prime power order

## DOI:

https://doi.org/10.11575/cdm.v8i2.62187## Keywords:

Math, Discrete, Cayley graphs, integral graphs, circulant graphs, graph energy## Abstract

The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs,

which can be characterized by their vertex count $n$ and a set $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z},\, \gcd(a-b,n)\in {\cal D}\}$.

Given an arbitrary prime power $p^s$, we determine all divisor sets maximising the energy of an integral circulant graph of order $p^s$. This enables us to compute the maximal energy $\Emax{p^s}$ among all integral circulant graphs of order $p^s$.

## References

O. Ahmadi and N. Alon and I.F. Blake and I.E. Shparlinski,

Graphs with integral spectrum,

Linear Algebra Appl. {\bf 430} (2009), 547-552.

------------------------------------------------

\bibitem{AKH} R. Akhtar and M. Boggess and T. Jackson-Henderson and I. Jim{\'e}nez and R. Karpman and A. Kinzel and D. Pritikin,

On the unitary Cayley graph of a finite ring,

Electron. J. Combin. {\bf 16} (2009), Research Paper R117, 13 pp. (electronic).

--------------------------------------------------

\bibitem{BAP} R.B. Bapat and S. Pati,

Energy of a graph is never an odd integer,

Bull. Kerala Math. Assoc. {\bf 1} (2004), 129-132.

--------------------------------------------------

\bibitem{BAS3} M. Ba\v{s}i\'c and M.D. Petkovi\'c,

{Perfect state transfer in integral circulant graphs of non-square-free order},

Linear Algebra Appl. {\bf 433} (2010), 149-163

----------------------------------------------------

\bibitem{BEA} N. de Beaudrap,

{On restricted unitary {C}ayley graphs and symplectic transformations modulo {$n$}},

Electron. J. Combin. {\bf 17} (2010), Research Paper R69, 26 pp. (electronic).

-----------------------------------------------------

\bibitem{BER} P. Berrizbeitia and R. E. Giudici,

{On cycles in the sequence of unitary Cayley graphs},

Discrete Math. {\bf 282} (2004), 239-243.

----------------------------------------------------

\bibitem{BRU} R.A. Brualdi,

Energy of a graph, AIM Workshop Notes, 2006.

------------------------------------------------------

\bibitem{DAV} P. J. Davis,

{Circulant matrices},

John Wiley \&\ Sons, New York-Chichester-Brisbane, 1979.

-----------------------------------------------------

\bibitem{DEJ} I. J. Dejter and R. E. Giudici,

{On unitary Cayley graphs},

J. Combin. Math. Combin. Comput. {\bf 18} (1995), 121-124.

------------------------------------------------------

\bibitem{DRO} A. Droll,

{A classification of {R}amanujan unitary {C}ayley graphs},

Electron. J. Combin. {\bf 17} (2010), Research Note N29, 6 pp. (electronic).

------------------------------------------------------

\bibitem{GUT} I. Gutman,

{The energy of a graph},

Ber. Math.-Stat. Sekt. Forschungszent. Graz {\bf 103}, 1978.

--------------------------------------------------

\bibitem{ILI} A. Ili\'{c},

The energy of unitary Cayley graphs,

Linear Algebra Appl. {\bf 431} (2009), 1881-1889.

----------------------------------------------------

\bibitem{KIA} D. Kiani and M.M.H. Aghaei and Y. Meemark and B. Suntornpoch,

The energy of unitary Cayley graphs and gcd-graphs,

Linear Algebra Appl. {\bf 435} (2011), 1336-1343.

----------------------------------------------------

\bibitem{KLO} W. Klotz and T. Sander,

Some properties of unitary Cayley graphs,

Electron. J. Combin. {\bf 14} (2007), Research Paper R45, 12 pp. (electronic).

---------------------------------------------------

\bibitem{KLO2} W. Klotz and T. Sander,

{Integral {C}ayley graphs over abelian groups},

Electron. J. Combin. {\bf 17} (2010), Research Paper R81, 13 pp. (electronic).

----------------------------------------------------

\bibitem{KOO} J.H. Koolen and V. Moulton,

{Maximal energy graphs},

Adv. Appl. Math. {\bf 26} (2001), 47-52.

---------------------------------------------------

\bibitem{PET} M.D. Petkovi\'c and M. Ba\v{s}i\'c,

{Further results on the perfect state transfer in integral circulant graphs},

Comput. Math. Appl. {\bf 61} (2011), 300-312

----------------------------------------------------

\bibitem{PIR} S. Pirzada and I. Gutman,

Energy of a graph is never the square root of an odd integer,

Appl. Analysis and Discr. Math. {\bf 2} (2008), 118-121.

---------------------------------------------------

\bibitem{RAM} H.N. Ramaswamy and C.R. Veena,

{On the Energy of Unitary Cayley Graphs},

Electron. J. Combin. {\bf 16} (2009), Research Note N24, 8 pp. (electronic).

-----------------------------------------------

\bibitem{SA1} J.W. Sander and T. Sander,

{The energy of integral circulant graphs with prime power order},

Appl. Anal. Discrete Math. {\bf 5} (2011), 22-36.

--------------------------------------------------

\bibitem{SA2} J.W. Sander and T. Sander,

{Integral circulant graphs of prime power order with maximal energy},

Linear Algebra Appl. {\bf 435} (2011), 3212-3232.

------------------------------------------------

\bibitem{SA3} J.W. Sander and T. Sander,

{The maximal energy of classes of integral circulant graphs},

Discrete Appl. Math. {\bf 160} (2012), 2015-2029.

------------------------------------------------

\bibitem{SHP} I. Shparlinski,

{On the energy of some circulant graphs},

Linear Algebra Appl. {\bf 414} (2006), 378-382.

-----------------------------------------------

\bibitem{SO} W. So,

Integral circulant graphs,

Discrete Math. {\bf 306} (2005), 153-158.

## Downloads

## Published

## Issue

## Section

## License

This copyright statement was adapted from the statement for the University of Calgary Repository and from the statement for the Electronic Journal of Combinatorics (with permission).

The copyright policy for Contributions to Discrete Mathematics (CDM) is changed for all articles appearing in issues of the journal starting from Volume 15 Number 3.

Author(s) retain copyright over submissions published starting from Volume 15 number 3. When the author(s) indicate approval of the finalized version of the article provided by the technical editors of the journal and indicate approval, they grant to Contributions to Discrete Mathematics (CDM) a world-wide, irrevocable, royalty free, non-exclusive license as described below:

The author(s) grant to CDM the right to reproduce, translate (as defined below), and/or distribute the material, including the abstract, in print and electronic format, including but not limited to audio or video.

The author(s) agree that the journal may translate, without changing the content the material, to any medium or format for the purposes of preservation.

The author(s) also agree that the journal may keep more than one copy of the article for the purposes of security, back-up, and preservation.

In granting the journal this license the author(s) warrant that the work is their original work and that they have the right to grant the rights contained in this license.

The authors represent that the work does not, to the best of their knowledge, infringe upon anyoneâ€™s copyright.

If the work contains material for which the author(s) do not hold copyright, the author(s) represent that the unrestricted permission of the copyright holder(s) to grant CDM the rights required by this license has been obtained, and that such third-party owned material is clearly identified and acknowledged within the text or content of the work.

The author(s) agree to ensure, to the extent reasonably possible, that further publication of the Work, with the same or substantially the same content, will acknowledge prior publication in CDM.

The journal highly recommends that the work be published with a Creative Commons license. Unless otherwise arranged at the time the finalized version is approved and the licence granted with CDM, the work will appear with the CC-BY-ND logo. Here is the site to get more detail, and an excerpt from the site about the CC-BY-ND. https://creativecommons.org/licenses/

**Attribution-NoDerivsCC BY-ND**

This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.