\part{Advanced}


\chapter{Topological graph theory}
- Topological graph
- Whitney complex
- Graph embedding
- Graph structure theorem
- Crossing number


\chapter{Spectral graph theory}
Working on the spectrum (eigenvalues and eigenvectors) of adjacency matrix is a powerful technique for studying graphs.
\section{Kirchoff's theorem}



\chapter{Algebraic graph theory}
- Chromatic polynomial
- Chromatic number 
\section{4 color theorem}
\begin{theorem}[4 color theorem]
	\[ G \text{ is a loopless planar graph } \implies \chi(G) \leq 4\]
\end{theorem}



\chapter{Extremal graph theory}
Manten's theorem, interesting application of Cauchy-Schwarz inequality
Let $G$ be a triangle free graph
\[|E| \leq \frac{|V|^2}{4}\]