手性繞圈數(chiral winding number)Z與拓撲不變量(topological invariant) Z2皆為廣泛運用於探討一維週期材料電子結構拓撲性質(topological properties)的重要理論，能預測材料是否有受對稱性保護之邊界或邊緣能態(Boundary or Edge states)。但對於手性繞圈數與拓撲不變量的探討往往是透過能帶的貝里相位(Berry phase)或瓦尼爾函數(Wannier functions)等物理概念與數學運算推導，難以建立直觀的視覺圖像。
如極化現代理論(modern theory of polarization)中所述，材料能帶的貝里相位對應到該能帶瓦尼爾函數的平均位置，因此若是能利用圖論方法藉由材料的幾何結構直接推斷出瓦尼爾函數的分布，便能省略繁雜的拓撲參數計算直接理解材料電子結構的拓撲性質。
在本篇論文中，我們以化學結構式為基礎，提出邊界/邊緣態電子結構的圖論，以一維週期材料奈米石墨帶為研究系統為例，依據克拉六重態規則(Clar’s sextet rule)建構出奈米石墨烯緞帶的凱庫勒共振路易士結構，稱之為克拉結構，能在不進行複雜的拓撲參數計算下，就能直觀預測不同形狀的石墨烯材料，在材料邊緣自由電子的分布情況，並進一步預測不同材料，以不同方式相接時自由電子在邊界的分布狀態。接著並利用緊束縛近似模型(Tight Binding model)進行奈米石墨帶的微擾計算(perturbation method)，探討由克拉結構預測出來的自由電子在邊緣或邊界的穩定程度。
本篇還更進一步將此化學觀點延伸應用於傳統拓撲理論無法討論的金屬石墨帶材料系統，除了能圖像化的以克拉結構解釋金屬石墨帶的導電性，更解釋為何傳統拓撲參數於此類系統無法定義，並歸納整理出金屬材料在邊緣上的電子結構性質。
克拉結構可為手性繞圈數以及拓撲不變量提供簡單易懂之化學觀點，以直觀視覺圖像對照其計算原理與數學推導，除讓非相關領域的學者更容易理解，並確立了石墨稀材料邊緣或邊界能態與幾何結構的圖形關聯，用以簡化未來石墨烯材料的設計原則。

The chiral winding number Z and the topological invariant Z2 are important theoretical topological parameters in predicting the presence of symmetry-protected edge or boundary electronic states in one-dimensional periodic materials. The investigation of the chiral winding number and/or the topological invariant is usually approached through rigorous mathematical derivations of abstruse physical concepts including Berry phase, Wannier functions and so on, making it challenging to construct an intuitive visual understanding.
As described in the modern theory of polarization, the Berry phase of a band corresponds to the average position of the corresponding Wannier function. Therefore, if we can directly visualize the distribution of Wannier functions from the geometric structure of the material, it would allow for a direct understanding of the topological properties of the material's electronic structure without the complex calculation of topological parameters.
In this thesis, we introduce a chemical structural perspective and propose a graph theory for the electronic structure of the edge/boundary states. We apply the Kekulé resonance Lewis structure based on the Clar’s sextet rule, which is called Clar structure, on one-dimensional graphene nanoribbons which allows us to predict straightforwardly the distribution of free electrons at the edges of graphene materials with different shapes and sizes without the explicit derivation of theoretical topological parameters. Our theory also allows us to predict the distribution of free electrons at the boundaries when different materials are connected in different geometries. By employing the tight-binding approximation model, we then perform perturbation calculations on graphene nanoribbons to investigate the stability of the free electron distribution at the edges or boundaries predicted by the graph theory based on Clar structures.
Furthermore, we extend this chemical graph theory to the electronic structure of metallic graphene nanoribbons, for which the application of conventional topological theory becomes insufficient. In addition to providing a visual explanation for the conductivity of metallic graphene nanoribbons, we also explain why it is not possible to define the topological properties in these materials. Finally, we summarize and organize the electronic structure properties of metallic materials at the edge and boundary, revealing physical insight on their behavior.
The graph theory we proposed based on the Clar’s Sextet rule provides a comprehensible explanation and visual interpretations of the principles behind chiral winding number and topological invariant. It not only facilitates a straightforward rationalization for scholars from diverse fields but also establishes a graphical correlation between edge or boundary states in graphene materials and their geometric structure. This serves to simplify future design principles for graphene materials.

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