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The second least eigenvalue of the signless Laplacian of the complements of trees. (English) Zbl 1467.05150

Summary: Suppose that \(\mathfrak{T}_n^c\) is a set, such that the elements of \(\mathfrak{T}_n^c\) are the complements of trees of order \(n\). In 2012, Li and Wang gave the unique graph in the set \(\mathfrak{T}_n^c\setminus\{K_{1, n-1}\}\) with minimum 1st ‘least eigenvalue of the signless Laplacian’ (abbreviated to a LESL). In the present work, we give the unique graph with 2nd LESL in \(\mathfrak{T}_n^c\setminus\{K^c_{1,n-1}\}\), where \(K_{1,n-1}^c\) represents the complement of star of order \(n\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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