Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs II

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• 作者：Lu Qiao ^{water6qiao@mail.nwpu.edu.cn} ; Shenggui Zhang ; ^{sgzhang@nwpu.edu.cn} ; Jing Li ^{jingli@nwpu.edu.cn}
• 关键词：Coulson integral formula ; General Laplacian energy-like invariant ; Graph energy
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：449
• 期：2
• 页码：1725-1740
• 全文大小：350 K
• 卷排序：449

文摘

Let G be a graph of order n   and λ1≥λ2≥⋯≥λnλ1≥λ2≥⋯≥λn be the eigenvalues of G. The energy of G   is defined as E(G)=∑k=1n|λk|. A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ1≥μ2≥⋯≥μn=0μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The general Laplacian energy-like invariant of G  , denoted by LELα(G)LELα(G), is defined as ∑μk≠0μkα when μ1≠0μ1≠0, and 0 when μ1=0μ1=0, where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α   is an irrational number where 0<|α|<10<|α|<1, whereas they do not hold when |α|>1|α|>1.