On the Skew Spectral Moments of Trees and Unicyclic Graphs

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 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.