The {graph} of a polytope is formed by the vertices and edges of the polytope.We often see this graph as an abstract graph and apply methodology from graph theory. A {polytopal graph} is simply a graph of a polytope. Appendix C reviews the relevant graph-theoretical prerequisites. Graphs of 3-polytopes are planar, and thus we review the topological background to study graphs embedded in a topological space. We then study properties of polytopal graphs. We analyse acyclic orientations of graphs of polytopes in Section 3.5; these orientations are related to the shelling orders of the corresponding dual polytopes. We also examine convex realisations of 3-connected planar graphs and Steinitz’s characterisation of graphs of 3-polytopes. The graph of a 3-polytope contains a subdivision of the graph of the 3-simplex, namely $K^{4}$.Section 3.9 shows that this extends to every dimension. Since $K^{5}$ is the 1-skeleton of a $4$-simplex, the nonplanarity of $K^{5}$ is a special case of a theoremof Flores (1934) and Van Kampen (1932) that states the $d$-skeleton of the $(2d+2)$-simplex cannot be embedded in $\R^{2d}$ (Section 3.10).

A nontrivial graph is \textit {$r$-connected}, for $r\ge 0$, if it has more than $r$ vertices and no two vertices are separated by fewer than $r$ other vertices. Graphs of $d$-polytopes are $d$-connected, according to Balinski (1961). The chapter also discusses a recent result of Pilaud et al (2022) on the edge connectivity of simplicial polytopes. We examine the higher connectivity of strongly connected complexes in Section 4.3. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs.Graphs of $d$-polytopes are $\floor{(d+2)/3}$-linked, but not all graphs of $d$-polytopes are $\floor{d/2}$-linked.Edge linkedness can be defined similarly by replacing vertex-disjoint paths in the definition of linkedness by edge-disjoint paths. The $d$-connectivity of a graph of a $d$-polytope is a particular case of a more general result of Athanasiadis (2009) on the connectivity of $(r,r+1)$-incidence graphs of a $d$-polytope. The chapter ends with a short discussion on the connectivity of incidence graphs.