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The algebraic connectivity of a graph G is the second-smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. The magnitude of this value reflects how well connected the overall graph is.

for an example, "adding self-loops" does not change laplacian eigenvalues (specially algebraic connectivity) of graph. Because, laplacian(G)= D-A is invariant with respect to adding self-loops.

My question is:

Does anyone has studied effect of different operations (such as edge contraction) on spectrum of laplacian? do you know good references?

Remark: the exact definition of the algebraic connectivity depends on the type of Laplacian used. For this question I prefer to use Fan Chung definition in SPECTRAL GRAPH THEORY. In this book Fan Chung has uesed a rescaled version of the Laplacian, eliminating the dependence on the number of vertices.

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    $\begingroup$ It would help if you provide some motivation and background. Please see How to ask a good question? and the site's FAQ. $\endgroup$
    – Kaveh
    Commented Mar 11, 2011 at 21:54
  • $\begingroup$ I'm also interested in the edge contraction case. I've spend some time previously trying to find references about the relation between eigenvalues and minor operations, with no success. $\endgroup$
    – Hsien-Chih Chang 張顯之
    Commented Mar 16, 2011 at 1:33
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    $\begingroup$ To me, the motivation seems fairly clear. $\endgroup$
    – Suresh Venkat
    Commented Sep 24, 2011 at 20:12
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    $\begingroup$ I second Suresh, knowing how various operations influence the Laplacian is interesting in itself and this problems shows up in various contexts. $\endgroup$
    – Marcin Kotowski
    Commented Sep 24, 2011 at 20:50

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Intuitively operations that preserve connectivity will not decrease the eigenvalues. For example, adding edges to the graph does not decrease the connectivity.

In general, if H is a subgraph of a graph G, by interlacing we know that the i-th largest Laplacian eigenvalue of H is no larger than the i-th largest Laplacian eigenvalue of G. A proof can be found in Proposition 3.2.1 of the book "Spectra of graphs" by Brouwer and Haemers. Note that the definition of Laplacian used in the book is not normalized; it has node degrees on the diagonal and -1 (or 0 if there is no edge) in the off-diagonal entries.

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  • $\begingroup$ Thanks Chang. Your answer is really useful for me. But if we use the definition of Laplacian that is not normalized, then many of comparisons seems to be meaningless. For example we have Algebraic Connectivity(K10)=10 and Algebraic Connectivity(K20)=20. however, both graphs are fully connected simple graphs. But if we use the normalized Laplacian, then NormalizedAlgebraicConnectivity(K10) = NormalizedAlgebraicConnectivity(K20)=1 and therefore comparison of normalized version seems to be more rational and natural. $\endgroup$
    – j.s.
    Commented Mar 16, 2011 at 2:03
  • $\begingroup$ @behnam: I agree with you. But after the normalization, some of the nondecreasing properties may differ. (Say one can ensure a strict decreasing on the largest Laplacian when deleting edges for the unnormalized ones, but not for the normalized ones.) $\endgroup$
    – Hsien-Chih Chang 張顯之
    Commented Mar 16, 2011 at 17:16
  • $\begingroup$ @Hsien-Chih Chang "Intuitively operations that preserve connectivity will not decrease the eigenvalues. For example, adding edges to the graph does not decrease the connectivity." Are you sure? Do you have a proof? Is the following a counter example? Start with a path graph and add edges to form the Lollipop graph. The cover time gets worse; it goes from $\Theta(n^2)$ to $\Theta(n^3)$. What is happening to the eigenvalues in this example? $\endgroup$
    – Tyson Williams
    Commented Sep 25, 2011 at 13:36
  • $\begingroup$ @TysonWilliams You are very right. This is one of the theoretical aspects where normalized and unnormalized Laplacians do differ. By adding edges, the unnormalized algebraic connectivity always goes up (Fiedler), but there are (not even very tricky) examples showing that edges added in the wrong places might diminish connectivity. This is particularly easy to see if you allow for edge weights. $\endgroup$
    – Delio M.
    Commented Oct 9, 2016 at 12:47

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