Study on Properties of Divisor Graphs of Finite Groups

Abstract

Algebraic graph theory provides a natural bridge between algebra and graph theory by encoding algebraic relationships as adjacency structures. Among these, divisor graphs and coprime graphs represent two contrasting perspectives on integer interactions. In this paper, we investigate the structural properties of the divisor graph whose vertices correspond to the integers graph invariants of and where adjacency is defined by divisibility. Fundamental , including connectivity, diameter, vertex degree distribution, and planarity, are examined in detail. We prove that is connected for all , establish an upper bound on its diameter, identify vertex as the unique maximum-degree vertex, and characterize the precise range of for which is planar.