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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si1.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=9c3c98df1650017b32da8e319b5420f3">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">$E(G)={\sum }_{k=1}^n|{\lambda }_k|$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si2.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=61e0f8dfb13f6c4284da01a311ec3a74" title="Click to view the MathML source">λ₁,λ₂,…,λ_nclass="mathContainer hidden">class="mathCode">${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si22.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=cfbdec6fee3c76ad064c854b5fe1c20f" title="Click to view the MathML source">μ₁≥μ₂≥⋯≥μ_n=0class="mathContainer hidden">class="mathCode">${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n=0$ be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G  , denoted by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si4.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=b77363dc48f51eb5f774d95069a55d3d" title="Click to view the MathML source">LEL_α(G)class="mathContainer hidden">class="mathCode">${\mathit{LEL}}_{\alpha }(G)$, is defined as class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si5.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=2db054da11d64a04c72f66e3b9575450">class="imgLazyJSB inlineImage" 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">class="mathContainer hidden">class="mathCode">${\sum }_{{\mu }_k\ne 0}{\mu }_k^{\alpha }$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si6.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=82f47e6dfa6881f81613c2b466cb6cea" title="Click to view the MathML source">μ₁≠0class="mathContainer hidden">class="mathCode">${\mu }_1\ne 0$, and 0 when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si7.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=d6d4803d2e84b3b790e324a39737acf9" title="Click to view the MathML source">μ₁=0class="mathContainer hidden">class="mathCode">${\mu }_1=0$, where α   is a real number. In this paper we give a Coulson-type integral formula for the general Laplacian-energy-like invariant for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si148.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=1182fedffcf03e7c367f645753f110c8" title="Click to view the MathML source">α=1/pclass="mathContainer hidden">class="mathCode">$\alpha =1/p$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15009981&_mathId=si149.gif&_user=111111111&_pii=S0022247X15009981&_rdoc=1&_issn=0022247X&md5=bb17c2b31707f2d88adc052f77e392cc" title="Click to view the MathML source">p∈Z⁺\{1}class="mathContainer hidden">class="mathCode">$p\in {\mathbb{Z}}^{+}\backslash \{1\}$. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.