Path Energy Bounds For Hub-Centric Graphs Of Order 2n+1 With Algorithmic Computation
This study investigates the path energy bounds of hub-centric graph families of order (2n+1), denoted by ℋ2n+1. The path energy is defined as the absolute sum of the eigenvalues of the path adjacency matrix Ap(ℋ2n+1), where each entry pij measures the maximum number of internally vertex- disjoint paths. Through an in-depth examination of the characteristic values of this matrix Ap(ℋ2n+1), we derive path energy bounds specific to these four (2n+1)-vertex graph, namely closed helm, double star, friendship, and double wheel graphs. We further developed an explicit spectral construction enabling the numerical validation algorithm to evaluate the time complexity in various structural configurations of (2n+1)-graphs. Also executed extensive trials to capture their average, maximum, and minimum computational performances. This analysis offers a detailed comparative study of the structural attributes that lead to computational complexity, highlighting the substantial algorithmic demands for several graphs.