# 2.8 Degrees & Isomorphism

## Extreme Graphs

For a simple connected graph with a total degree of 44, what is

1. the minimum number of vertices?

Exercise 1

A complete graph minimizes the number of vertices. $$K_7$$ has a total degree of 42. Adding on a vertex and two edges achieves a total degree of 44.

2. the maximum number of vertices?

Exercise 2

A line graph maximizes the number of vertices: the first and the last vertices both have degree 1, while the vertices in-between have degree 2. (44+2)/2 = 23, so the line graph $$L_{23}$$ has a total degree of 44.