CS 598: Spectral Graph Theory: Lecture 3 - Corelab

Courant-Fischer for Laplacian
• Applying Courant-Fischer for the Laplacian
we get :
2
max
min
x1,
x0
1
0,
v 1
1
T
x Lx
T
x x
min
x1,
x0
T
x Lx
max max
x0
T
x x x0
S of dim k
• Useful for getting bounds, if calculating spectra is cumbersome.
• To get upper bound on λ2, just need to produce vector with small
Rayleigh Quotient.
• Similarly, t o get lower bound on λmax, just need to produce vector
with large Rayleigh Quotient
( i,
j)
E
( i,
j)
E
( x
iV
( x
iV
i
i
x
x
2
i
x
x
2
i
j
j
)
)
2
2
k
min
max
xS
T
x Lx
T
x x