Analyzing Pathway Energy And Time Complexity In Flower Families Through Vertex-Disjoint Paths

This research delves into the pathway energy framework for flower families, a class of simple connected graphs, whose path matrix p is constructed such that each entry pij quantifies the maximum number of vertex-disjoint paths. By analyzing the characteristic values of this matrix, we establish the pathway energy bounds specific to these flower graph families. Additionally, a comprehensive algorithm is developed to evaluate the time complexity across different flower family configurations, utilizing numerous trials to capture their average, maximum, and minimum computational behaviors. This analysis offers a comparative study of the structural intricacies that lead to increased computational complexity, highlighting which graph topologies tend to impose higher algorithmic challenges. The proposed method introduces a refined and adaptable approach, deepening the exploration of characteristic graph properties and their computational impact, thereby expanding the practical applications of these findings in graph theory.