On the skew eigenvalues of joined union of oriented graphs and applications (Q6622109)

By the joined union $G$ of oriented graphs $G_i$ for $1\leq i\leq n$, we mean the union of oriented graphs $G_1,\dots,G_n$ together with the arcs $(v_ik, v_jl)$, where $v_ik\in G_i$ and $v_jl\in G_j$, whenever $(v_i, v_j)$ is an arc in $G$. The authors of this paper show that the skew eigenvalues of the joined union of oriented graphs are the union of the skew eigenvalues of the component oriented graphs except for some eigenvalues, which are given by an auxiliary matrix associated with the joined union. They also obtain as a special case the skew eigenvalues of the join of two oriented graphs and the lexicographic product of oriented graphs. Their results extend and generalize the results obtained by \textit{H. S. Ramane} et al. [Trans. Comb. 5, No. 1, 15--23 (2016; Zbl 1463.05228)].