A path in a graph encompassing its whole vertex set is called Hamiltonian. Such a path with sharing the same initial and terminal vertices is called a Hamiltonian cycle. A graph comprising a Hamiltonian path (resp. cycle) is said to be traceable (resp. Hamiltonian). Graphs possessing Hamiltonian paths between every pair of their vertices are said to be Hamilton-connected. The computational complexity of evaluating a graph to be Hamilton-connected is NP-complete. A detour is the longest path in a graph. The detour index is the sum of the length of detours between every unordered pair of vertices. Computing the detour index of a graph is an NP-complete problem as well. A finite subset $P\subset {\mathbb{R}}^{\epsilon }$ is called a convex polytope if $P$ is a convex hull. In this paper, we devised two distinct methods to prove a graph to be Hamilton-connected and employed these methods to construct some infinite families of Hamilton-connected convex polytopes. The convex polytope $B_{\epsilon }$ has been shown to be non-Hamilton-connected in the literature. We showed that the existing proof for $B_{\epsilon }$ is false and showed that this family is, in fact, Hamilton-connected. The paper is concluded with study implications followed by some future directions.

In this study, we focus on the Padmakar-Ivan (PI) index, a molecular descriptor that quantifies the structural characteristics of chemical networks on the basis of their vertex distances. The main objective of this work is to develop efficient approaches for computing the PI index, particularly for graph powers and complex network structures. We begin by formulating an algorithm for computing the PI index of general graphs and extend it to the $k^{\text{th}}$ power of a graph using a distance-based framework. Furthermore, we introduce a novel clique cut method that establishes a theoretical foundation for analyzing and computing the PI index in intricate silicate and silicon-based frameworks. The proposed techniques significantly simplify and generalize existing computational procedures.

Line graphs are a fundamental class of graphs extensively studied for their structural properties and applications in diverse fields such as network design, optimization, and algorithm development. Pan and lollipop graphs, with their distinctive hybrid structures, offer a fertile ground for exploring combinatorial properties in their line graphs. Motivated by the need to better understand domination, chromaticity, and Hamiltonian properties in line graphs, this study examined the line graphs of pan and lollipop graphs. These investigations were inspired by their potential applications in connectivity analysis and optimization in networks. We derived analytical formulas for the domination and chromatic numbers of these line graphs, established relationships between these parameters and their corresponding original graphs, and proved that the line graph of a pan graph is Hamiltonian while that of a lollipop graph is traceable. The methodology combines established theoretical results and inequalities, including domination bounds and chromaticity relations, with rigorous combinatorial analysis. Our results not only contribute to the theoretical understanding of line graphs but also have implications for practical problems in network optimization and graph algorithm design, opening avenues for further research into hybrid graph structures.