**Transcribed Image Text:**

## Example 2.1.3 Let's calculate the adjacency matrix for the graph from Example 1.7.1 drawn again.
below. We consider the vertices in alphabetical order. Since there are edges between A and B, A and
C, and A and E, in the first row for A we have a 1 in the 2nd, 3rd, and 5th column and Os elsewhere.
B
E
D
A
B
C
D
E
ABCDE
011011
101 10
1 10 10
01100
10000
4. (a) Write the adjacency matrix for the complete graph K, and describe the general pattern for
the adjacency matrix of K₁, for n 2 1.
(b). Write the adjacency matrix for the complete bipartite graph K₁3 and describe the general
pattern for the adjacency matrix of Km for n. m 2 1.
5. (a) Write the adjacency matrix for cycle C₁ and describe the general pattern for the adjacency
matrix of C₁ for n 2 3.
(b) Write the adjacency matrices for wheel W, and explain how to get the adjacency matrix of
W₁, using the matrix for Cn.
Hint: see how the matrix of W4 includes the matriz of C₁?
A graph is regular if all of its vertices have the same degree. If a graph is regular with all vertices of
degree r, we say the graph is r-regular. For example, the 4-cycle C4 is 2-regular whereas the complete
graph K4 is 3-regular.
6. Which of these graphs are regular? Explain. Note: it might depend on n or m.
(a) Cycle Cn
(b) Path P
(c) Complete Kn
(d) Bipartite Km,n