Laplacian Eigenvalue Trace Sums

Problem

Return the sum and the sum of squares of the Laplacian eigenvalues without computing eigenvalues.

The trace of L equals the sum of eigenvalues and trace(L^2) equals the sum of squared eigenvalues; build L = D - A and evaluate both by matrix multiplication. edges is a list of [u, v] pairs. Return [trace, trace_square] as integers.

Hint

trace(L^2) also equals sum of squared degrees plus twice the number of edges.

Test cases

case 1

in: [3,[[0,1],[1,2]]]

expected: [4,10]

case 2

in: [3,[[0,1],[1,2],[2,0]]]

expected: [6,18]

case 3

in: [4,[[0,1],[0,2],[0,3]]]

expected: [6,18]

case 4

in: [1,[]]

expected: [0,0]

case 5

in: [4,[[0,1],[1,2],[2,3],[3,0]]]

expected: [8,24]

Starter codePython

def laplacian_spectrum_sums(n, edges):
    # Your code here
    pass

5 test cases · run your code against them in the editor