The eigenvalues and energy of integral circulant graphs

Document Type : Research Paper


Amirkabir University


‎A graph is called \textit{circulant} if it is a Cayley graph on a‎ ‎cyclic group‎, ‎i.e‎. ‎its adjacency matrix is circulant‎. ‎Let $D$ be a‎ ‎set of positive‎, ‎proper divisors of the integer $n>1$‎. ‎The‎ ‎integral circulant graph $ICG_{n}(D)$ has the vertex set‎ ‎$\mathbb{Z}_{n}$ and the edge set E$(ICG_{n}(D))= \{\{a,b\};‎ ‎gcd(a-b,n)\in D \}$‎. ‎Let $n=p_{1}p_{2}\cdots p_{k}m$‎, ‎where‎ ‎$p_{1},p_{2},\cdots,p_{k}$ are distinct prime numbers and‎ ‎$gcd(p_{1}p_{2}\cdots p_{k},m)=1$‎. ‎The open problem posed in paper‎ ‎[A‎. ‎Ili'{c}‎, ‎The energy of unitary Cayley graphs‎, ‎Linear Algebra‎ ‎Appl.‎, ‎431 (2009) 1881--1889] about calculating the energy of an‎ ‎arbitrary integral circulant $ICG_{n}(D)$ is completely solved in‎ ‎this paper‎, ‎where $D=\{p_{1},p_{2},\ldots,p_{k} \}$‎.


Main Subjects

R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jim\'{e}nez, R. Karpman, A. Kinzel and D. Pritikin (2009). On the unitary Cayley graph of a finite ring. Electron. J. Combin., Research Paper 117. 16 S. Blackburn and I. Shparlinski (2008). On the average energy of circulant graphs. Linear Algebra Appl.. 428, 1956-1963 R. A. Brualdi Energy of a Graph. \href{}{}. D. Cvetkovi$\rm\acute{c}$, M. Doob and H. Sachs (1980). Spectra of Graphs. Academic Press, New York. D. M. Cvetkovi$\rm\acute{c}$, P. Rowlinson and S. Simi$\rm\acute{c}$ (1997). Eigenspaces of Graphs. Cambridge University Press, Cambridge. C. Godsil and G. Royle (2001). Algebraic Graph Theory. Springer-Verlag, New York. I. Gutman (1978). The energy of a graph. Ber. Math. Stat. Sekt. Forschungszent. Graz.. 103, 1-22 I. Gutman (2001). The energy of a graph: old and new results. In: A. Betten, A. Kohnert, R. Laue and A. Wassermann, Editors, Algebraic Combinatorics and Applications, Springer-Verlag, Berlin. , 196-211 I. Gutman, D. Kiani, M. Mirzakhah and B. Zhou (2009). On incidence energy of a graph. Linear Algebra Appl.. 431, 1223-1233 F. K. Hwang (2003). A survey on multi-loop networks. Theor. Comput. Sci.. 299, 107-121 A. Ili$\rm\acute{c}$ (2009). The energy of unitary Cayley graphs. Linear Algebra Appl.. 431, 1881-1889 A. Ili$\rm\acute{c}$ (2010). Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl.. 433, 1005-1014 A. Ili$\rm\acute{c}$ and M. Ba$\rm\check{s}$i$\rm\acute{c}$ (2010). On the chromatic number of integral circulant graphs. Comp. Math. Appl.. 60, 144-150 A. Ili$\rm\acute{c}$ and M. Ba$\rm\check{s}$i$\rm\acute{c}$ (2011). New results on the energy of integral circulant graphs. Appl. Math. Comput.. 218, 3470-3482 M. R. Jooyandeh, D. Kiani and M. Mirzakhah (2009). Incidence energy of a graph. MATCH Commun. Math. Comput. Chem.. 62, 561-572 D. Kiani, M. Molla Haji Aghaei, Y. Meemark and B. Suntornpoch (2011). Energy of unitary Cayley graphs and gcd-graphs. Linear Algebra Appl.. 435, 1336-1343 D. Kiani and M. Molla Haji Aghaei (2012). On the unitary Cayley graphs of a ring. Electron. J. Combin., P10.. 19 (2) W. Klotz and T. Sander (2007). Some properties of unitary Cayley graphs. Electron. J. Combin., Reasearch paper 45.. 14 H. N. Ramaswamy and C. R. Veena (2009). On the Energy of Unitary Cayley Graphs. Electron. J. Combin., Note 24.. 16 W. So (2006). Integral circulant graphs. Discrete Math.. 306, 153-158 J. W. Sander and T. Sander (2011). The energy of integral circulant graphs with prime power order. 5, 22-36
Volume 1, Issue 3 - Serial Number 3
September 2012
Pages 47-56
  • Receive Date: 10 October 2012
  • Revise Date: 16 October 2012
  • Accept Date: 16 October 2012
  • Published Online: 01 September 2012