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

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• 作者：Lu Qiao ^{water6qiao@mail.nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shenggui Zhang ; ^{sgzhang@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bo Ning ^{ningbo_math84@mail.nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jing Li ^{jingli@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Laplacian matrix ; Coulson integral formula ; General Laplacian-energy-like invariant
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1249-1261
• 全文大小：335 K

文摘

Let G   be a simple graph. Its energy is defined as lineImage" height="18" width="130" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15009981-si1.gif">$E(G)={\sum }_{k=1}^n|{\lambda }_k|$, where lick to view the MathML source">λ₁,λ₂,&hellip;,λ_n${\lambda }_1,{\lambda }_2,\&hel<font\; color="red">li</font>p;,{\lambda }_n$ are the eigenvalues of G  . A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let lick to view the MathML source">μ₁≥μ₂≥⋯≥μ_n=0${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n=0$ be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G  , denoted by lick to view the MathML source">LEL_α(G), is defined as lineImage" height="20" width="72" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15009981-si5.gif">${\sum }_{{\mu }_k\ne 0}{\mu }_k^{\alpha }$ when lick to view the MathML source">μ₁≠0${\mu }_1\ne 0$, and 0 when lick to view the MathML source">μ₁=0${\mu }_1=0$, where α   is a real number. In this paper we give a Coulson-type integral formula for the general Laplacian-energy-like invariant for lick to view the MathML source">α=1/p$\alpha =1/p$ with lick to view the MathML source">p∈Z⁺\{1}$p\in {\mathbb{Z}}^{+}\backslash \{1\}$. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.