On the Inverse Problem for Some Topological Indices
Durbar Maji, Ganesh Ghorai, Muhammad Khalid Mahmood, Md. Ashraful Alam
Abstract
The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>g</a:mi> </a:mrow> </a:mfenced> </a:math> to every graph within a given family <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mi mathvariant="script">G</f:mi> </f:mrow> </f:mfenced> </f:math> of graphs, find some <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M3"> <l:mi>G</l:mi> <l:mo>∈</l:mo> <l:mi mathvariant="script">G</l:mi> </l:math> for which <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M4"> <o:mtext>TI</o:mtext> <o:mfenced open="(" close=")" separators="|"> <o:mrow> <o:mi>G</o:mi> </o:mrow> </o:mfenced> <o:mo>=</o:mo> <o:mi>g</o:mi> </o:math> . It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M5"> <t:mi>Y</t:mi> </t:math> -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M6"> <v:mi>F</v:mi> </v:math> -index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.