A Matrix Approach to Vertex-Degree-Based Topological Indices

A VDB (vertex-degree-based) topological index over a set of digraphs H is a function φ : H → R , defined for each H ∈ H as φ H = 1 2 ∑ u v ∈ E φ d u + d v − , where E is the arc set of H , d u + and d v − denote the out-degree and in-degree of vertices u and v respectively, and φ i j = f ( i , j ) for an appropriate real symmetric bivariate function f . It is our goal in this article to introduce a new approach where we base the concept of VDB topological index on the space of real matrices instead of the space of symmetric real functions of two variables. We represent a digraph H by the p × p matrix α H , where α H i j is the number of arcs u v such that d u + = i and d v − = j , and p is the maximum value of the in-degrees and out-degrees of H . By fixing a p × p matrix φ , a VDB topological index of H is defined as the trace of the matrix φ T α ( H ) . We show that this definition coincides with the previous one when φ is a symmetric matrix. This approach allows considering nonsymmetric matrices, which extends the concept of a VDB topological index to nonsymmetric bivariate functions.