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(2) Graph Connectivity Recall that the eigenvalues of the Laplacian of a graph on

`n`

vertices can be ordered as

`0=\lambda _(1)<=\lambda _(2)<=cdots<=\lambda _(n)`

. Consider the following graph

`H`

: (a) Write down the Laplacian

`L_(H)`

of the graph

`H`

. (b) Compute the quadratic function

`x^(TT)L_(H)x`

and write it as a sum of squares. (c) Find a basis for the eigenspace of

`L_(H)`

corresponding to the eigenvalue 0 . (d) Based on the above which is the first eigenvalue of

`L_(H)`

that is going to be positive? (e) What is the relationship between the arithmetic multiplicity of the eigenvalue 0 and the number of connected components in a graph? Explain.