Find the adjacency matrix of the intersection of W4 and K5. What is the degree of vertexes in W4 K5.

Considering the usual definition of intersection of graphs, i.e. the graph we get by taking the intersection of all vertices and the intersection of all edges, is dependent on the labelling of vertices we would usually need more information.

However both graphs have 5 vertices and K5 is isomorphic under all labelings of vertices, for any labelling of vertices the adjacency matrix of K5 is:

01111
10111
11011
11101
11110

Lets label W4 such that the center vertex is v1 and the remaining ones are v2 through v5 where v2 and v4 are not adjacent and v3 and v5 are not adjacent. Then the adjacency matrix of W4 is:

01111
10101
11010
10101
11010

Finally we find the adjacency matrix of the intersection, so vi and vj are adjacent in the intersection if and only if they are adjacent in both K5 and W4. This means the coefficient a_ij in the adjacency matrix of the intersection is 1 iff a_ij is 1 in both the adjacency matrix of K5 and W4. The adjacency matrix of the intersection is therefore:

01111
10101
11010
10101
11010

The same as the adjacency matrix of W4. Finally we can read off the degree of each vertex by taking the sum of the coefficients in each row, the degree of v1 (the center vertex) is 4 and the degree of the rest of the vertices is 3.