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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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