## Noisy random graphs and their laplacians.(English)Zbl 1153.05062

This paper deals with the spectra of weighted graphs. Define a Wigner-noise matrix to be an $$n\times n$$ matrix $$W$$ whose entries $$w_{ij}$$ are uniformly bounded random variables with expectations zero and variances bounded by some $${\sigma^2}$$, informally, a small noisy perturbation matrix. The eigenvalues of such $$W$$ are known to be \ small\' with probability tending to 1, in a precise sense, by a result of Füredi and Komlós. The author then considers the effect of replacing a weighted adjacency matrix $$B$$ of some $$n$$-vertex graph $$G$$ by $$A=W+B$$ where $$W$$ is a Wigner-noise matrix for certain particular choices of weighted graph $$G$$ with $$n$$ vertices and weighted adjacency matrix $$B$$.
At first, $$B$$ is a Kronecker-sum of $$n_{i}\times n_{i}$$ matrices $$B_{i}$$ (where $$\sum_{i=1}^{k}n_{i}=n$$) and each $$B_{i}$$ has all off-diagonal entries equal to $$\mu_{i}$$ and all diagonal entries equal to $$\nu_{i}$$. (Thus $$G$$ has $$k$$ components, each pair of vertices connected with an edge of the same weight). Somewhat more generally, $$B$$ has a so-called pattern matrix: this means that $$V(G)=V_{1}\cup V_{2}\cup \ldots \cup V_{k}$$ where $$| V_{i}|=n_{i}$$ and all edge weights between a vertex in $$V_{i}$$ and one in $$V_{j}$$ have weight $$p_{ij}$$: here $$P=(p_{ij})$$ is the $$k\times k$$ pattern matrix. $$B$$ is then called a blown-up matrix. For both cases, the author has proven in earlier papers that there are $$k$$ \ protruding\' eigenvalues of order $$\Theta(n)$$ of $$B$$ with the others being smaller, provided the so-called growth condition holds, namely that each $$n_{i}\geq Cn$$.
Section 2 of the paper uses linear algebra methods to give an alternative proof that, under the growth rate condition, all non-zero eigenvalues of the $$n\times n$$ blown-up matrix $$B$$ are order $$n$$ in absolute value. This then leads to the fact that the protruding eigenvalues of $$A$$ can be found and the block structure of $$B$$ can be read off from the eigenvectors of $$A$$.
The author also considers weighted Laplacians $$L^{\prime}(A)$$, (recall the Laplacian is $$D-A$$ where $$D$$ is a diagonal matrix with diagonal entries the degrees of the vertices: then $$L^{\prime}(A)=I_{n}-D(A)^{-1/2}AD(A)^{-1/2}$$ (if no vertex is isolated). She shows that, under the growth rate condition, there are $$k$$ eigenvalues of $$L^{\prime}(B)$$ which are not equal to 1 and they are located in the union of intervals $$[0,1-\delta]\cup [1+\delta,2]$$ for a constant $$\delta\in (0,1)$$: the proof is again linear algebra. Again this leads to the statement of a similar result for $$L^{\prime}(A)$$, which it is said will be proved in a later paper. Again eigenvectors of $$A$$ will reveal the block structure in $$B$$.
The author closes by discussing links with the Szemerédi Regularity Lemma briefly.

### MSC:

 05C80 Random graphs (graph-theoretic aspects) 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A42 Inequalities involving eigenvalues and eigenvectors
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### References:

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