In recent times, there has been a significant interest in low-dimensional materials due totheir unique electronic, optical, magnetic, and topological properties that differ from 3D bulk materials. This dissertation focuses on a specific class of 1D carbon structures known as graphene nanoribbons (GNRs), which can be synthesized atomically with precision through a bottom-up method. The theoretical tools employed in this study are primarily topological theory and quantum many-body first-principles calculations. Chapter 1 introduces some basics about density functional theory, GW many-body perturbation theory and the Belthe-Salpeter equation. Chapter 2 of this dissertation delves into the topology of GNRs when chiral symmetry is approximately maintained. Building on this theory and in collaboration with experimentalists, Chapter 3 explores a metallic 1D nanowire known as saw-tooth GNRs, while Chapter 4 investigates various quantum dot systems with unique bonding and anti-bonding characters. In Chapter 5, a different type of metallic GNRs is studied using zero-mode (topologically protected in-gap electronic states) engineering. Chapter 6 takes the study beyond the Hermitian Hamiltonian and introduces the non-Hermitian skin effect. When 1D or 0D structures are interconnected, nanoporous graphene is formed. Its electronic properties are studied in Chapter 8. Furthermore, Chapter 9 examines a carbon kagome lattice’s excitonic properties. The content of each Chapter is elaborated as the following: • Chapter 1 provides a foundational understanding of density functional theory (DFT) for ground state properties by introducing the Kohn-Sham equation and different functionals. We also discuss the GW perturbation theory, which allows us to incorporate many-body effects into our calculations of excited-state properties. Specifically, we explore how the GW method can be utilized to calculate quasi-particle excitations. To study the two-particle excitation problem for optical properties, we introduce the Bethe-Salpeter equation (BSE) method. This equation provides a framework for calcu2 lating the interaction between an excited electron and the hole it leaves behind, which is crucial for understanding optical properties such as absorption and emission spectra. • In Chapter 2, we examine GNR structures under the first nearest neighbor tightbinding model, assuming chiral symmetry holds. In this scenario, we utilize the first Chern number to obtain a Z index for general 1D materials. From the general Z index formula, we derive the Chiral phase index in vector form, which enables us to obtain the analytic Z index formula for all types of unit cells in GNRs. Finally, we explore a spin-chain formed by topological junction states that exhibit strong spin-spin interactions.[1] • Chapter 3 builds on the chiral classification theory introduced in Chapter 2 by utilizing the topological junction states as building blocks and connecting them in a symmetric manner to form a 1D metallic nanowire. We use first-principles DFT calculations to study the electronic bandstructure, local density of states (LDOS), and mapping of wavefunctions. Our results are then compared with experimental STM measurements, and we achieve good agreement. In addition, we also investigate the topological properties of asymmetrically connected structures, and the predicted junction/end state matches well the corresponding experimental evidence.[2] • In Chapter 4, we employed the topological junction states that arise from the connection between 7-armchair graphene nanoribbons (7AGNR) and 9-armchair graphene nanoribbons (9AGNR) to construct topological quantum dots. We investigated two distinct types of quantum dots by means of DFT calculations, with the aim of studying their electronic properties, such as the bonding and anti-bonding traits of their valence and conduction states. In addition, we devised a tight-binding theory to elucidate the underlying factors contributing to the characteristics of the wavefunctions.[3] • In Chapter 5, we focus on a different variety of metallic graphene nanoribbon (GNR) called Olympicene GNRs that does not exhibit the Stoner instability, which was observed in the sawtooth GNRs presented in Chapter 3. This new GNR features coveshaped edges, and its low-energy behavior is governed by zero modes. The most notable distinction between this GNR and the sawtooth GNR is that the nearest zero modes localize on different sublattices, leading to a significant increase in electron hopping and precluding any magnetic instability. To verify this, we conduct DFT calculations and compare our findings with experimental observations. • In Chapter 6, we explore the topology of 1D non-Hermitian systems, extending our analysis beyond Hermitian topological classification. Specifically, we investigate a 1D non-Hermitian system with no symmetry constraints, and use a Z index that can be employed to classify such systems. We examine the well-known skin effect for non-trivial non-Hermitian topological models and identify a promising GNR material, Co-4AGNR, which could potentially be realized in experiments. By conducting firstprinciples DFT and full-frequency GW calculations, we establish that the material 3 exhibits non-trivial topology. Lastly, we present evidence of the asymmetric transport properties in this material by calculating the Green’s function for a finite segment of this system. • In Chapter 7, we examine the 2D carbon structure that results from linking 1D metallic GNRs. To accomplish this, we created a theoretical model with low energy states using modes that are found in the pentagons located at the edge of GNRs as the bases. This effective tight-binding model provides a description of a unique, distorted super-graphene. We also conducted DFT calculations and compared our findings with experimental results provided by our colleagues.[4] • Chapter 8 focuses on the examination of a kagome lattice that is formed by linking triangulene building blocks. This unique structure was predicted to exhibit excitonic insulator (EI) behavior. In partnership with experimentalists, we conducted an investigation of the electronic properties of this structure using multiple levels of theory, such as DFT, GW-BSE, and Bardeen-Cooper-Schrieffer (BCS) theory. Our research revealed that DFT based single-particle theory was insufficient for accurately capturing the features of the LDOS map observed in STM measurements. By incorporating a BCS-like theory for condensation of excitons, we were able to provide an explanation for the experimental observations.[5] In addition to the projects above, I was also involved in 3 other projects, including one studying the color center in twisted BN [6], one studying the kondo effect in magnetic Ndoped chevronGNR [7], one studying the pseodo-atomic orbitals in graphene nanoribbons [8]. These research projects are also very interesting, but beyond the scope of this dissertation.