Theory of generalized splines

Let $R$ be a commutative ring with identity and $G=(V,E)$ be a finite graph, where $V$ is a set of elements of vertices and $E$ is a set of elements of edges. A map $ \alpha : E \to \{\text{ideals in R}\}$ is called \emph{an edge-labeling function}, which label edges of $G$ by nonzero ideals of $R$. The pair $(G,\alpha)$ is called an \emph{edge-labeled graph}. \emph{A generalized spline} is a vertex labeling $F \in R^{\mid V \mid}$ on an edge-labeled graph $(G,\alpha)$ so that the difference between labels of any two adjacent vertices lies in the corresponding edge ideal. The collection of all generalized splines on $(G,\alpha)$ is denoted by $R_{(G, \alpha)}.$ It has a ring and a $R$- module structure. In this thesis, we study over $R$-module structure on generalized splines and we find a flow-up minimum generating set for $[\mathbb{Z}/ m\mathbb{Z}]_{(K_n,\alpha)}$. \hfil \vspace{3mm}\\ This thesis includes four chapters. In Chapter 1, we give a survey of the literature on classical spline theory and generalized spline theory. \hfil \vspace{3mm} \\ In Chapter 2, we give the necessary background knowledge of some definitions and theorems of ring, module and graph theory. \hfil \vspace{3mm} \\ In Chapter 3, we introduce generalized spline theory and give some properties. We especially focus on generalized spline modules on complete graphs. Furthermore, we define flow- up classes, which is a special type of generalized splines, and give some properties. \hfil \vspace{3mm} \\ In Chapter 4, we study two different methods to find a flow-up minimum generating set for complete graphs over $\mathbb{Z}/ m\mathbb{Z}$. For the first method, we use some previous works done by Philbin and others [\ref{2017}]. We first obtain a flow-up minimum generating set for $[\mathbb{Z}/ p^k\mathbb{Z}]_{(K_n,\alpha)}$ by using the algorithm they developed over $\mathbb{Z}/ p^k\mathbb{Z}$ (see Algorithm \ref{algoritma}). Then, we examine some examples in order to find a flow-up minimum generating set for $[\mathbb{Z}/ m\mathbb{Z}]_{(K_n,\alpha)}$ by using the algorithm they developed over $\mathbb{Z}/ m\mathbb{Z}$ (see Algorithm \ref{algm}). We observe that depending on the magnitude of $\mid V \mid$ or $m$, it is quite difficult to find a flow-up minimum generating set for $[\mathbb{Z}/ m\mathbb{Z}]_{(K_n,\alpha)}$. Therefore, we give another method to find a flow-up minimum generating set for complete graphs over $\mathbb{Z}/ m\mathbb{Z}$. We first modify some results of Altınok and Sarıoğlan [\ref{2019}] over $\mathbb{Z}/ m\mathbb{Z}$ in order to use them in our works. Then, we obtain a flow-up minimum generating set for complete graphs over $\mathbb{Z} / m \mathbb{Z}$.