On the Skew Spectral Moments of Trees and Unicyclic Graphs 
Received:May 05, 2022 Revised:October 04, 2022 
Key Words:
oriented graph skew spectral moment tree unicyclic graph

Fund Project:Supported by the Research Project of Jianghan University (Grant No.2021yb056) and the National Natural Science Foundation of China (Grant Nos.11971158; 12061039). 

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Abstract: 
Given a simple graph $G$, the oriented graph $G^\sigma$ is obtained from $G$ by orienting each edge and $G$ is called the underlying graph of $G^\sigma$. The skewsymmetric adjacency matrix $S(G^\sigma)$ of $G^\sigma$, where the $(u,v)$entry is $1$ if there is an arc from $u$ to $v$, and $1$ if there is an arc from $v$ to $u$ (and 0 otherwise), has eigenvalues of 0 or pure imaginary. The $k$thskew spectral moment of $G^\sigma$ is the sum of power $k$ of all eigenvalues of $S(G^\sigma)$, where $k$ is a nonnegative integer. The skew spectral moments can be used to produce graph catalogues. In this paper, we researched the skew spectral moments of some oriented trees and oriented unicyclic graphs and produced their catalogues in lexicographical order. We determined the last $2\lfloor\frac{d}{4}\rfloor$ oriented trees with underlying graph of diameter $d$ and the last $2\lfloor\frac{g}{4}\rfloor+1$ oriented unicyclic graphs with underlying graph of girth $g$, respectively. 
Citation: 
DOI:10.3770/j.issn:20952651.2023.04.002 
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