On the Eigenvalues of Rhomboidal C4C8(R)[n; n] Nanotori

Document Type: Research Paper


Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran


A C4C8 net is a trivalent decoration made by alternating squares C4 and octagons C8. It can
cover either a cylinder or a torus. In this paper, we study the adjacency spectrum of rhomboidal
C4C8 tori. We also give lower and upper bounds for a chemical quantity, namely Estrada index,
for a C4C8 net.


Main Subjects

1. F. Afashari and M. Maghasedi, Rhomboidal c4c8 tori which are Cayley
Graphs, arXiv:1807.00278v1.
2. B. Alspach and M. Dean, Honeycomb toroidal graphs are Cayley graphs,
Inform. Process. Lett. 109 (2009) 705−708.
3. A. Cayley, The theory of groups: Graphical representations, Amer. J. Math.
1 (1878) 174−176.
4. D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs − Theory and
Applications, third ed., Johann Ambroisus Barth Verlag, Heidelberg,
Leipzig, 1995.

5. M. DeVos, L. Goddyn, B. Mohar and R. Samal, Cayley sum graphs and
eigenvalues of (3,6)-fullerenes, J. Combin. Theory 99 (2009) 358−369.
6. P. Diaconis and M. Shahshahani, Generating a random permutation with
random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981) 159−179.
7. M. V. Diudea and P. E John, Covering polyhedral tori, MATCH Commun.
Math. Comput. Chem. 44 (2001) 103−116.
8. M. V. Diudea, B. Parv, P. E. John, O. Ursu and A. Graovac, Distance
counting in tori, MATCH Commun. Math. Comput. Chem. 49 (2003)
9. J. D. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in
Mathematics 163, Springer−Verlag, New York, 1996.
10. E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett.
319 (2000) 713−718.
11. E. Estrada, Characterization of the folding degree of proteins,
Bioinformatics 18 (2002) 697−704.
12. C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in
Mathematics 207, Springer−Verlag, New York, 2001.
13. I. Gutman, S. Radenković, A. Graovac and D. Plavsic, Monte Carlo
approach to Estrada index, Chem. Phys. Lett. 446 (2007) 233−236.
14. I. Gutman and S. Radenković, A lower bound for the Estrada index of
bipartite molecular graphs, Kragujevac J. Sci. 29 (2007) 67−72.
15. P. E. John and H. Sachs, Spectra of toroidal graphs, Discrete Math. 309
(2009) 2663−2681.
16. P. Kaski, Eigenvectors and Spectra of Cayley Graphs, Accompanying
manuscript to the seminar presentation given in "T−79.300 Postgraduate
Course in Theoretical Computer Scince", Helsiniki University of
Technology, Spring term, 2002.
17. W. Lederman, Introduction to Group Characters, second edition,
Cambridge University Press, Cambridge, 1987.
18. N. J. Rad and A. Jahanbani, D. A. Mojdeh, Tetracyclic graphs with
maximal Estrada index, Discrete Math. Alg. Appl. 9 (3) (2017) 1750041.
19. G. Sabidussi, On a class of fixed-point-free graphs, Proc. Amer. Math. Soc.
9 (1958) 800−804.
20. J. P. Serre, Linear Representations of Finite Groups, Graduate Texts in
Mathematics 42, Springer−Verlag, New York, 1977.