WORKED EXAMPLE 1.5
A graph has 3 vertices and 6 edges.
a What is the sum of the degrees of the vertices? b Explain why the graph must have loops or multiple edges. c Draw three different graphs that fit this description. a
edges; sum is twice the number of edges.
6 Ă— 2 = 12
6
b If there are no loops or multiple edges, then the
PL E
maximum degree at each vertex is 2, giving degree sum = 6 . The degree sum is greater than 6, so there must be loops or multiple edges.
With no loops or multiple edges, the maximum degree sum is 6.
c For example:
M
The first graph shown has multiple edges; the second has multiple edges and a loop; the third is made up of two disconnected graphs and has multiple edges and loops.
Key point 1.7
A subgraph of a graph is formed by using some or all of the vertices of a graph
SA
together with some or all of the edges that connect these vertices. A subgraph is a graph contained within another graph. This could result in an unconnected vertex. However, subgraphs are usually connected.
Key point 1.8
Subdivision means inserting a vertex of degree 2 into an edge. Subdivision increases the number of vertices by 1 and the number of edges by 1.
WORKED EXAMPLE 1.6
Original material Š Cambridge University Press 2018