On a conjecture for the sum of Laplacian eigenvalues
On a conjecture for the sum of Laplacian eigenvalues
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DOI：
10.1016/j.mcm.2011.12.047
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发表时间：
2012-08
期刊：
影响因子：
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通讯作者：
Shouzhong Wang;Yufei Huang;Bolian Liu
中科院分区：
文献类型：
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作者：
Shouzhong Wang;Yufei Huang;Bolian Liu
设G是一个有n个顶点和e(G)条边的简单图。browwer等人推测G的k个最大拉普拉斯特征值之和不超过e(G)+k+12，其中1≤k≤n。本文证明了k足够大的连通图、单环图、双环图、带约束的三环图、森林等情况下，这个猜想是成立的。进一步证明了如果G是一个具有特定性质的树，那么G的k个最大拉普拉斯特征值的和不超过e(G)+2k−2，其中1≤k≤n。
Let G be a simple graph with n vertices and e(G) edges. Brouwer et al. conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G)+k+12, where 1≤k≤n. In this paper, this conjecture is proved to be true for the following cases: connected graphs with sufficiently large k, unicyclic graphs, bicyclic graphs and tricyclic graphs with some restrictions, forests, etc. Moreover, we show that if G is a tree with a specified property, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k−2, where 1≤k≤n.