This set is due Monday, April 29, at the beginning of lecture.
- Suppose is an matrix with complex entries such that . Verify that for real.
- Suppose that is nilpotent. Verify that is invertible.
- Use linear algebra to find a closed form expression for the terms of the sequence given by the recurrence relation , , for all .
- (Norman Biggs, Algebraic Graph Theory, Proposition 2.3) Let be a (finite, simple) graph with vertices, and let be its adjacency matrix. Suppose that the characteristic polynomial of is . Verify that:
- is the number of edges in .
- is twice the number of triangles in .