A Matrix Approach to Vertex-Degree-Based Topological Indices

Abstract

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 φH=12∑uv∈Eφdu+dv−, where E is the arc set of H, du+ and dv− 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 αH, where αHij is the number of arcs uv such that du+=i and dv−=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 φTα(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.

Keywords

• VDB topological indices
• digraphs
• space of matrices
• general first Zagreb index