Remarks on the Graovac-Ghorbani index of bipartite graphs

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• 作者：Darko Dimitrov^a ; ^{darko.dimitrov11@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Barbara Ikica ; ^b ; ^c ; ^{barbara.ikica@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author} ; ^{barbara.ikica@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Riste &Scaron ; krekovski^b ; ^d ; ^e ; ^{skrekovski@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Molecular structure descriptor ; Molecular graph ; Extremal graphs ; Atom&ndash ; bond connectivity index ; Graovac&ndash ; Ghorbani index
• 刊名：Applied Mathematics and Computation
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：293
• 期：Complete
• 页码：370-376
• 全文大小：443 K

文摘

The atom–bond connectivity (ABC) index is a well-known degree-based molecular structure descriptor with a variety of chemical applications. In 2010 Graovac and Ghorbani introduced a distance-based analog of this index, the Graovac–Ghorbani (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$) index, which yielded promising results when compared to analogous descriptors. In this paper, we investigate the structure of graphs that maximize and minimize the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$ index. Specifically, we show that amongst all bipartite graphs, the minimum class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$ index is attained by a complete bipartite graph, while the maximum class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$ index is attained by a path or a cycle-like graph; the structure of the resulting graph depends on the number of vertices. Through the course of the research, we also derive an asymptotic estimate of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$ index of paths. In order to obtain our results, we introduce a normalized version of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si4.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=c0f2ae54e68a4107e9d115a881665e80" title="Click to view the MathML source">GGclass="mathContainer hidden">class="mathCode">$\mathrm{GG}$ index and call it the normalized Graovac–Ghorbani (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300316305422&_mathId=si5.gif&_user=111111111&_pii=S0096300316305422&_rdoc=1&_issn=00963003&md5=17e8c0e90b0d6aa2743f273153544739" title="Click to view the MathML source">NGGclass="mathContainer hidden">class="mathCode">$\mathrm{NGG}$) index. Finally, we discuss some related open questions as a potential extension of our work.