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).
 Author Name Affiliation Yaping WU School of Artificial Intelligence, Jianghan University, Hubei 430056, P. R. China Huiqing LIU School of Mathematics and Statistics, Hubei University, Hubei 430062, P. R. China Qiong FAN School of Mathematics and Statistics, Central China Normal University, Hubei 430079, P. R. China
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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 skew-symmetric 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 $k$-th-skew spectral moment of $G^\sigma$ is the sum of power $k$ of all eigenvalues of $S(G^\sigma)$, where $k$ is a non-negative 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.