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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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