# On the skew spectral moments of trees with a given bipartition

Document Type : Research Paper

Authors

1 School of Artificial Intelligence, Jianghan University, 430056, Wuhan, P. R.China

2 School of Mathematics and Statistics, Central China Normal University, 430070, Wuhan, P. R.China

3 School of Mathematics and Statistics, Hubei University, 430062, Wuhan, P. R.China

Abstract

Let $G$ be a simple graph, and $\vec{G}$ be an oriented graph of $G$ with an orientation and skew-adjacency matrix $S(\vec{G})$. Let $\lambda_1(\vec{G}), \lambda_2(\vec{G}),\ldots,\lambda_n(\vec{G})$ be the eigenvalues of $S(\vec{G})$. The number $\sum_{i=1}^{n}\lambda_i^k(\vec{G})$ $(k=0, 1,\ldots,n-1)$, denoted by $T_k(\vec{G})$, is called the $k$-th {\em skew spectral moment} of $\vec{G}$, and $T(\vec{G})=(T_0(\vec{G}),T_1(\vec{G}),\ldots,$ $T_{n-1}(\vec{G}))$ is the sequence of skew spectral moments of $\vec{G}$. Suppose $\vec{G}_1$ and $\vec{G}_2$ are two digraphs. We shall write $\vec{G}_1\prec_T \vec{G}_2$ ($\vec{G}_1$ comes before $\vec{G}_2$ in a $T$-order) if for some $k$ $(1 \leq k \leq n-1)$, $T_i(\vec{G}_1)=T_i(\vec{G}_2)$ ($i=0, 1,\ldots,k-1$) and $T_k(\vec{G_1})< T_k(\vec{G}_2)$ hold. For two given positive integers $p$ and $q$ with $p \leq q$, we denote $\mathscr T_{n}^{p,q}=\{T: T$ is a tree of order $n$ with a $(p,q)$-bipartition $\}$. In this paper, we discuss $T$-order among all trees in $\mathscr T_{n}^{p,q}$. Furthermore, the last three trees, in the $T$-order, underlying graphs among $\mathscr T_{n}^{p,q}~(4\leq p\leq q)$ are characterized.

Keywords

Main Subjects

#### References

[1] J. A. Bondy and U. S. R. Murty, Graph theory with applications, American Elsevier Publishing Co., Inc., New York,
1976.
[2] M. Cavers, S. M. Cioabă, 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] D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs Theory and Applications, Academic Press, New York, 1980.
[4] D. Cvetković, M. Doob, H. Sachs and A. Torgas̆ev, Recent results in the theory of graph spectra, Annals of Discrete
Mathematics, 36, North-Holland Publishing Co., Amsterdam, 1988.
[5] D. Cvetković and P. Rowlinson, Spectra of unicyclic graphs, Graphs Combin., 3 (1987) 7–23.
[6] S. C. Li and J. J. Zhang, Lexicographical ordering by spectral moments of trees with a given bipartition, Bull.
Iranian Math. Soc., 40 (2014) 1027–1045.
[7] S. Gong and G. Xu, 3-Regular digraphs with optimum skew energy, Linear Algebra Appl., 436 (2012) 465–471.
[8] Y. Hou and T. Lei, Charteristic polynominal of skew-adjacency matrices of orinted graphs, Electron.J.Combin., 18
(2011) 156–167.
[9] B. Shader and W. So, Skew spectra of oriented graphs, Electron. J. Combn., 16 (2009) 1–6.
[10] F. Taghvaee and G. H. Fath-Tabar, Signless Laplacian spectral moments of graphs and ordering some graphs with respect to them, Alg. Struc. Appl., 1 (2014) 133–141.
[11] F. Taghvaee and G. H. Fath-Tabar, On the skew spectral moments of graphs, Transactions on Combin., 1 (2017)
47–54.
[12] Y. P. Wu, H. Q. Liu and Q. Fan, On the spectral moments of signless Laplacian matrix of trees and unicyclic graphs, Ars Combin., 141 (2018) 345–351.

### History

• Receive Date: 18 May 2021
• Revise Date: 08 February 2023
• Accept Date: 11 February 2023
• Published Online: 01 June 2024