Extremal Problems for Forests in Graphs and Hypergraphs

The Turan number, ex_r(n; F), of an r-uniform hypergraph F is the maximum number of hyperedges in an n-vertex r-uniform hypergraph which does not contain F as a subhypergraph. Note that when r = 2, ex_r(n; F) = ex(n; F) which is the Turan number of graph F. We study. Turan numbers in the degenerate case for graphs and hypergraphs; we focus on the case when F is a forest in graphs and hypergraph. In the first chapter we discuss the history of Turan numbers and give several classical results. In the second chapter, we examine the Turan number for forests with path components, forests of path and star components, and forests with all components of order 5. In the third chapter we determine the Turan number of an r-uniform star forest in various hypergraph settings

On the Connectivity of KK-distance Graphs

For any $k \in \mathbb{N}$, the $k-$distance graph $D^{k}G$ has the same vertex set of $G$, and two vertices of $D^{k}G$ are adjacent if they are exactly distance $k$ apart in the original graph $G$. In this paper, we consider the connectivity of $D^{k}G$ and state the conditions for graph $G$ and integer $k$ such that the graph $D^{k}G$ is connected