On Topological Indices for New Classes of Benes Network
Aftab Hussain, Muhammad Numan, Nafisa Naz, Saad Ihsan Butt, Adnan Aslam, Asfand Fahad
Abstract
Topological indices (TIs) transform a molecular graph into a number. The TIs are a vital tool for quantitative structure activity relationship (QSAR) and quantity structure property relationship (QSPR). In this paper, we constructed two classes of Benes network: horizontal cylindrical Benes network <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mtext>HCB</a:mtext> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>r</a:mi> </a:mrow> </a:mfenced> </a:math> and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. We derive analytical close formulas for general Randić connectivity index, general Zagreb, first and the second Zagreb (and multiplicative Zagreb), general sum connectivity, atom-bond connectivity ( <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mtext>VCB</f:mtext> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mi>r</f:mi> </f:mrow> </f:mfenced> </f:math> ), and geometric arithmetic <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"> <k:mfenced open="(" close=")" separators="|"> <k:mrow> <k:mtext>ABC</k:mtext> </k:mrow> </k:mfenced> </k:math> index of the two classes of Benes networks. Also, the fourth version of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M4"> <p:mtext>GA</p:mtext> </p:math> and the fifth version of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M5"> <r:mtext>ABC</r:mtext> </r:math> indices are computed for these classes of networks.