Energy of a digraph with respect to a VDB topological index

Abstract

Abstract Let DD be a digraph with vertex set VV and arc set EE. For a vertex uu, the out-degree and in-degree of uu are denoted by du+{d}_{u}^{+} and du−{d}_{u}^{-}, respectively. A vertex-degree-based (VDB) topological index φ\varphi is defined for DD as φ(D)=12∑uv∈Eφdu+,dv−,\varphi (D)=\frac{1}{2}\sum _{uv\in E}{\varphi }_{{d}_{u}^{+},{d}_{v}^{-}}, where φi,j{\varphi }_{i,j} is an appropriate function which satisfies φi,j=φj,i{\varphi }_{i,j}={\varphi }_{j,i}. In this work, we introduce the energy ℰφ(D){{\mathcal{ {\mathcal E} }}}_{\varphi }(D) of a digraph DD with respect to a general VDB topological index φ\varphi , and after comparing it with the energy of the underlying graph of its splitting digraph, we derive upper and lower bounds for ℰφ{{\mathcal{ {\mathcal E} }}}_{\varphi } and characterize the digraphs which attain these bounds.