The inextricable links among adjacency matrix, incidence matrix, and topological indices

Abstract

Graph matrices provide a concise and structured representation of molecules, facilitating the application of graph theory in the field of molecular chemistry. In this research, we used three types of graph matrices: the adjacency matrix, the incidence matrix, and the distance matrix, to establish the inherited relationship between a few topological indices and their associated graph matrices. Also, we derived a new formula to count the number of triangles in a simple graph using the singular values of graph matrices. The sum of all elements of a matrix $Su\left(A\right)$ plays a vital role in establishing these relations. For this study, both distance-based and degree-based indices were taken into account. We explored the topological indices of the cartesian product of a graph and the subdivision graphs of a given graph using graph matrices.