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">, 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,…,λnclass="mathContainer hidden">class="mathCode"> 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≥⋯≥μn=0class="mathContainer hidden">class="mathCode"> 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">LELα(G)class="mathContainer hidden">class="mathCode">, 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"> 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">, 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">, 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"> 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">. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.