STRUCTURAL CHARACTERIZATION OF RECURSIVE PENTAGONAL SHELL GRAPHS USING RECURSIVE GROWTH PARAMETERS

Abstract

Recursive graph families are essential in graph theory because of their hierarchical structure and recursive growth patterns. In this paper, a new planar graph family called the Recursive Pentagonal Shell Graph (RPSG) is developed and built from a triangular seed graph using sequential recursive expansions. Each cycle adds three new vertices, resulting in an extra pentagonal shell while keeping the previous graph structure. The recursive construction is explained using vertex labelling, adjacency representation and recursive growth rules. To characterize the structural behaviour of the suggested graph, three new factors are introduced namely the Pentagonal Shell Index, Expansion Rate and Outer Boundary Size. Several structural features such as vertex development, recursive shell generation, connectedness, planarity and boundary preservation are supported by strong mathematical proofs. The suggested framework offers a systematic approach to analyse recursively layered planar graphs and serves as the foundation for future research into recursive graph families and their structural features.