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On the bounds of Laplacian eigenvalues of k -connected graphs

Xiaodan ChenYaoping Hou — 2015

Czechoslovak Mathematical Journal

Let μ n - 1 ( G ) be the algebraic connectivity, and let μ 1 ( G ) be the Laplacian spectral radius of a k -connected graph G with n vertices and m edges. In this paper, we prove that μ n - 1 ( G ) 2 n k 2 ( n ( n - 1 ) - 2 m ) ( n + k - 2 ) + 2 k 2 , with equality if and only if G is the complete graph K n or K n - e . Moreover, if G is non-regular, then μ 1 ( G ) < 2 Δ - 2 ( n Δ - 2 m ) k 2 2 ( n Δ - 2 m ) ( n 2 - 2 n + 2 k ) + n k 2 , where Δ stands for the maximum degree of G . Remark that in some cases, these two inequalities improve some previously known results.

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