Topological Indices for the Identification of Breast Cancer: A Mathematical Perspective

Abstract

In this paper, the graph G=(V,E) is considered a simple, connected graph. Topological indices, which provide numerical descriptors of graph structures, have recently gained attention as potential diagnostic tools in medical studies. In graph-theoretical modeling, the structure of healthy breast tissue can be represented as a composition of equal-sized square lattice graphs. For such regular structures, the calculation of topological indices, particularly the first and second Zagreb indices, is straightforward, and their behavior follows a well-defined geometric pattern. In contrast, pathological breast tissue exhibits structural irregularities and increased complexity, which can be modeled using lattice-like graphs with non-uniform, disordered configurations. This study aims to propose four graph models representing pathological breast tissue and to compare them with the healthy tissue model by computing the first and second Zagreb indices. Results demonstrate that the values of these indices for pathological graph models differ significantly from those of the healthy graph model. Consequently, any substantial deviation in the values of the first or second Zagreb index for lattice graphs corresponding to healthy breast tissue may indicate possible pathological alterations, thereby highlighting the necessity for clinical examination.