On the skew spectral moments of graphs

Document Type : Research Paper

Authors

University of Kashan

Abstract

Let $G$ be a simple graph‎, ‎and $G^{\sigma}$‎ ‎be an oriented graph of $G$ with the orientation ‎$\sigma$ and skew-adjacency matrix $S(G^{\sigma})$‎. ‎The $k-$th skew spectral‎ ‎moment of $G^{\sigma}$‎, ‎denoted by‎ ‎$T_k(G^{\sigma})$‎, ‎is defined as $\sum_{i=1}^{n}( ‎‎‎\lambda_{i})^{k}$‎, ‎where $\lambda_{1}‎, ‎\lambda_{2},\cdots‎, ‎\lambda_{n}$ are the eigenvalues of $G^{\sigma}$‎. ‎Suppose‎ ‎$G^{\sigma_1}_{1}$ and $G^{\sigma_2}_{2}$ are two digraphs‎. ‎If there‎ ‎exists an integer $k$‎, ‎$1 \leq k \leq n-1$‎, ‎such that for each‎ ‎$i$‎, ‎$0 \leq i \leq k-1$‎, ‎$T_i(G^{\sigma_1}_{1}) =‎ ‎T_i(G^{\sigma_2}_{2})$ and‎ ‎$T_k(G^{\sigma_1}_{1}) <T_k(G^{\sigma_ 2}_{2})$‎ ‎then we write‎ ‎$G^{\sigma_1}_{1} \prec_{T} G^{\sigma_2}_{2}$‎.
‎In this paper‎, ‎we determine some of the skew spectral moments of oriented graphs‎. ‎Also we order some oriented unicyclic graphs with respect to skew spectral moment‎.

Keywords

Main Subjects

References

[1] A. R. Ashra and G. H. Fath-Tabar, Bounds on the Estrada index of ISR (4,6)-fullerenes, Appl. Math. Lett., 24 (2011) 337-339.
[2] M. Cavers, S. M. Cioaba, S. Fallat, D. A. Gregory, W. H. Haemers, S. j. Kirkland, J. J. McDonald and M.
Tsatsomeros, Skew adjacency matrices of graphs, Linear Algebra Appl., 436 (2012) 5412-5429.
[3] X. Chen, X. Li and H. Lian, 4-Regular oriented graphs with optimum skew energy, Linear Algebra Appl., 439 (2013) 2948{2960.
[4] D. Cvetkovic and P. Rowlinson, Sp ectra of unicyclic graphs, Graphs Combin., 3 (1987) 7-23.
[5] D. Cvetkovic, M. Do ob and H. Sachs, Spectra of Graphs-Theory and Applications, 87, Academic Press, New York, 1980.
[6] G. H. Fath-Tabar, A. R. Ashra and I. Gutman, Note on Estrada and L   Estrada indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math., 139 (2009) 1-16.
[7] G. H. Fath-Tabar, A. R. Ashra and D. Stevanovic, Sp ectral Prop erties of Fullerenes, J. Comput. Theor. Nanosci., 9 (2012) 327-329.
[8] S. Gong and G. Xu, 3-Regular digraphs with optimum skew energy, Linear Algebra Appl., 436 (2012) 465-471.
[9] Y. Hou and T. Lei, Characteristic p olynomials of skew-adjacency matrices of oriented graphs, Electron. J. Combin., 18 (2011) 156-167.
[10] B. Shader and W. So, Skew sp ectra of oriented graphs, Electron. J. Combin., 16 (2009) 1-6.
[11] F. Taghvaee and A. R. Ashra , Comparing fullerenes by sp ectral moments, J. Nanosci. Nanotechnol., 16 (2016) 1-4.
[12] F. Taghvaee and G. H. Fath-Tabar, Signless Laplacian sp ectral moments of graphs and ordering some graphs with resp ect to them, Alg. Struc. Appl., 1 (2014) 133-141.
[13] F. Taghvaee and G. H. Fath-Tabar, Relationship b etween co efficients of characteristic p olynomial and matching p olynomial of regular graphs and its applications, Iranian J. Math. Chem., 8 (2017) 7-24.
[14] Y. P. Wu and H. Q. Liu, Lexicographical ordering by sp ectral moments of trees with a prescrib ed diameter, Linear Algebra Appl., 433 (2010) 1707-1713.

Files

History

  • Receive Date: 10 February 2016
  • Revise Date: 03 August 2016
  • Accept Date: 08 September 2016
  • Published Online: 01 March 2017

Share

How to cite

Statistics