A Matrix Approach to Vertex-Degree-Based Topological Indices (2024)
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) =12 ∑uv∈Eφd+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 φij = 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)]ij is the number of arcs uv 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 Tα(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.