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.