A new notion of energy of digraphs

Document Type : Research Paper


School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China


The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. Let $z_1,\ldots,z_n$ be the eigenvalues of an $n$-vertex digraph $D$. Then we give a new notion of energy of digraphs defined by $E_p(D)=\sum_{k=1}^{n}|{\Re}(z_k) {\Im}(z_k)|$, where ${\Re}(z_k)$ (respectively, ${\Im}(z_k)$) is real (respectively, imaginary) part of $z_k$. We call it $p$-energy of the digraph $D$. We compute $p$-energy formulas for directed cycles. For $n\geq 12$, we show that $p$-energy of directed cycles increases monotonically with respect to their order. We find unicyclic digraphs with smallest and largest $p$-energy. We give counter examples to show that the $p$-energy of digraph does not possess increasing--property with respect to quasi-order relation over the set $\mathcal{D}_{n,h}$, where $\mathcal{D}_{n,h}$ is the set of $n$-vertex digraphs with cycles of length $h$. We find the upper bound for $p$-energy and give all those digraphs which attain this bound. Moreover, we construct few families of $p$-equienergetic digraphs.


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