In these notes, I tried to cover enough material to get a feeling for “calculus on graphs”, with applications to sports rankings and the Friendship Theorem. Here’s a list of the topics.

1 . **Introduction**

2. **Examples**

3. **Basic definitions**

3.1 Diameter, radius, and all that

3.2 Treks, trails, paths

3.3 Maps between graphs

3.4 Colorings

3.5 Transitivity

4. **Adjacency matrix**

4.1 Definition

4.2 Basic results

4.3 The spectrum

4.4 Application to the Friendship Theorem

4.5 Eigenvector centrality and the Keener ranking

4.6 Strongly regular graphs

4.7 Orientation on a graph

5. **Incidence matrix**

5.1 The unsigned incidence matrix

5.2 The oriented case

5.3 Cycle space and cut space

6. **Laplacian matrix**

6.1 The Laplacian spectrum

7 Hodge decomposition for graphs

7.1 Abstract simplicial complexes

7.2 The Bjorner complex and the Riemann hypothesis

7.3 Homology groups

8. **Comparison graphs**

8.1 Comparison matrices

8.2 HodgeRank

8.3 HodgeRank example