Energy of a digraph with respect to a VDB topological index

Abstract

Abstract Let D D be a digraph with vertex set V V and arc set E E . For a vertex u u , the out-degree and in-degree of u u are denoted by d u + {d}_{u}^{+} and d u − {d}_{u}^{-} , respectively. A vertex-degree-based (VDB) topological index φ \varphi is defined for D D as φ ( D ) = 1 2 ∑ u v ∈ E φ d u + , d v − , \varphi (D)=\frac{1}{2}\sum _{uv\in E}{\varphi }_{{d}_{u}^{+},{d}_{v}^{-}}, where φ i , j {\varphi }_{i,j} is an appropriate function which satisfies φ i , j = φ j , i {\varphi }_{i,j}={\varphi }_{j,i} . In this work, we introduce the energy ℰ φ ( D ) {{\mathcal{ {\mathcal E} }}}_{\varphi }(D) of a digraph D D 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 ℰ φ {{\mathcal{ {\mathcal E} }}}_{\varphi } and characterize the digraphs which attain these bounds.