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A domain monotonicity theorem for graphs and Hamiltonicity. (English) Zbl 0765.05071
The eigenvalues of the Laplacian matrix $$L=[L_{uv}]$$ of a graph $$(L_{vv}$$ is the degree of the vertex $$v$$ while $$L_{uv}$$ is minus the number of edges between vertices $$u$$ and $$v$$ for $$u\neq v)$$ are studied. Two theorems, which relate the eigenvalues of (the Laplacian matrix of) a graph and the eigenvalues of its induced subgraphs, are proved. A necessary condition for the existence of long cycles in a graph which involves the Laplacian spectrum is derived. As an example, it is proved, by such spectral methods, that the Petersen graph is not Hamiltonian.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs
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##### References:
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