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.

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Main Subjects


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