How to construct a k-regular graph?
If $k=2m$ is even, put all the vertices around a circle, and join each to its $m$ nearest neighbors on either side.
If $k=2m+1$ is odd, and $n$ is even, put the vertices on a circle, join each to its $m$ nearest neighbors on each side, and also to the vertex directly opposite.
Once you have an initial $k$-regular graph, you can generate many more by randomly applying the following simple switching operation:
provided it does not introduce a parallel edge or loop.
I just finished this question from Allen Downey's book Think Complexity, and thought I'll just share my algorithm (which is the same as Gerry's answer but just fleshed out) :
Preconditions: 0 < k and k < n
**If n is odd, k must be even
Number of Vertex (n) : Possible Number of degrees for each Vertex (k)
2 : 1
3 : 2
4 : 1,2,3
5 : 2,4
6 : 1,2,3,4,5
7 : 2,4,6
8 : 1,2,3,4,5,6,7
9 : 2,4,6,8
10 : 1,2,3,4,5,6,7,8,9
Examples:
if n is 7:
if k is 2:
make_edge_for_each_vertice_x_steps_away(1)
if k is 4:
make_edge_for_each_vertice_x_steps_away(1)
make_edge_for_each_vertice_x_steps_away(3)
if k is 6:
make_edge_for_each_vertice_x_steps_away(1)
make_edge_for_each_vertice_x_steps_away(3)
make_edge_for_each_vertice_x_steps_away(5)
if n is 8:
if k is 1:
make_edge_for_each_vertice_x_steps_away(4)
if k is 2:
make_edge_for_each_vertice_x_steps_away(1)
if k is 3:
make_edge_for_each_vertice_x_steps_away(4)
make_edge_for_each_vertice_x_steps_away(3)
if k is 4:
make_edge_for_each_vertice_x_steps_away(1)
make_edge_for_each_vertice_x_steps_away(2)
if k is 5:
make_edge_for_each_vertice_x_steps_away(4)
make_edge_for_each_vertice_x_steps_away(3)
make_edge_for_each_vertice_x_steps_away(2)
if k is 6:
make_edge_for_each_vertice_x_steps_away(1)
make_edge_for_each_vertice_x_steps_away(2)
make_edge_for_each_vertice_x_steps_away(3)
if k is 7:
make_edge_for_each_vertice_x_steps_away(4)
make_edge_for_each_vertice_x_steps_away(3)
make_edge_for_each_vertice_x_steps_away(2)
make_edge_for_each_vertice_x_steps_away(1)
make_edge_for_each_vertice_x_steps_away(x):
for each vertice, adds an edge to a neighbor x steps away.
if x == 1, edges will be created between v1 and v2, v2 and v3,..., vn to v1
There is a pattern where
-If n is odd, we add edges between vertices x positions away, where x is all odd numbers between 1 to k.
-If n is even, and k is even, we add edges between vertices x positions away, where x is a range of numbers starting from 1 to n/2, and the range of these numbers is limited by the number of even numbers from 1 to k, including k
-If n is even, and k is odd, we add edges between vertices x positions away, where x is a range of numbers starting from n/2 to 1, and the range of these numbers is limited by the number of odd numbers from 1 to k, including k
Algorithm:
Layout_all_vertices_in_a_circle()
if n is even:
if k is even:
countEvenNumbers = n/2
#add an edge for each even number between 1 to k, including k
for i in range(countEvenNumbers):
make_edge_for_each_vertice_x_steps_away(i+1)
else if k is odd:
countOddNumbers = ((n-1)/2) + 1
#add an edge for each odd number between 1 to k, including 1 and k
for i in range(countOddNumbers):
make_edge_for_each_vertice_x_steps_away((n/2)-i)
if n is odd:
for i in range(1, k):
if i is odd:
make_edge_for_each_vertice_x_steps_away(i)
Hope this helps!