A new notion of energy of digraphs

Abstract

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 $ngeq 12$, 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.