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A spectral lower bound for the treewidth of a graph and its consequences. (English) Zbl 1161.68647
Summary: We give a lower bound for the treewidth of a graph in terms of the second smallest eigenvalue of its Laplacian matrix. We use this lower bound to show that the treewidth of a $$d$$-dimensional hypercube is at least $$\lfloor 3\cdot 2d/(2(d+4))\rfloor - 1$$. The currently known upper bound is $$0(2^d/\sqrt d)$$. We generalize this result to Hamming graphs. We also observe that every graph $$G$$ on $$n$$ vertices, with maximum degree $$\Delta$$ contains an induced cycle (chordless cycle) of length at least $$1+\log_{\Delta}(\mu n/8)$$ (provided $$G$$ is not acyclic),has a clique minor $$K_h$$ for some $$h=\Omega ((n\mu ^{2}/(\Delta +2\mu )^{2})^{1/3})$$, where $$\mu$$ is the second smallest eigenvalue of the Laplacian matrix of $$G$$.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science
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##### References:
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