Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Abstract

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>G</a:mi> <a:mtext> </a:mtext> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>H</c:mi> </c:math> gives us a new graph obtained by taking one copy of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mi>G</e:mi> </e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"> <g:mfenced open="|" close="|"> <g:mrow> <g:mi>V</g:mi> <g:mfenced open="(" close=")"> <g:mrow> <g:mi>G</g:mi> </g:mrow> </g:mfenced> </g:mrow> </g:mfenced> </g:math> copies of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"> <m:mi>H</m:mi> </m:math> and joining the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M6"> <o:mi>i</o:mi> </o:math> th vertex of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M7"> <q:mi>G</q:mi> </q:math> to every vertex in the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M8"> <s:mi>i</s:mi> </s:math> th copy of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M9"> <u:mi>H</u:mi> </u:math> . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M10"> <w:msub> <w:mrow> <w:mi>P</w:mi> </w:mrow> <w:mrow> <w:mi>s</w:mi> </w:mrow> </w:msub> <w:mo>⨀</w:mo> <w:msub> <w:mrow> <w:mi>P</w:mi> </w:mrow> <w:mrow> <w:mi>t</w:mi> </w:mrow> </w:msub> </w:math> , <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M11"> <y:msub> <y:mrow> <y:mi>C</y:mi> </y:mrow> <y:mrow> <y:mi>t</y:mi> </y:mrow> </y:msub> <y:mo>⨀</y:mo> <y:msub> <y:mrow> <y:mi>C</y:mi> </y:mrow> <y:mrow> <y:mi>s</y:mi> </y:mrow> </y:msub> </y:math> , <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M12"> <ab:msub> <ab:mrow> <ab:mi>K</ab:mi> </ab:mrow> <ab:mrow> <ab:mi>t</ab:mi> </ab:mrow> </ab:msub> <ab:mo>⊙</ab:mo> <ab:msub> <ab:mrow> <ab:mi>K</ab:mi> </ab:mrow> <ab:mrow> <ab:mi>s</ab:mi> </ab:mrow> </ab:msub> </ab:math> , <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M13"> <cb:msub> <cb:mrow> <cb:mi>K</cb:mi> </cb:mrow> <cb:mrow> <cb:mi>t</cb:mi> </cb:mrow> </cb:msub> <cb:mo>⊙</cb:mo> <cb:msub> <cb:mrow> <cb:mi>P</cb:mi> </cb:mrow> <cb:mrow> <cb:mi>s</cb:mi> </cb:mrow> </cb:msub> </cb:math> , and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M14"> <eb:msub> <eb:mrow> <eb:mi>P</eb:mi> </eb:mrow> <eb:mrow> <eb:mi>s</eb:mi> </eb:mrow> </eb:msub> <eb:mo>⊙</eb:mo> <eb:msub> <eb:mrow> <eb:mi>K</eb:mi> </eb:mrow> <eb:mrow> <eb:mi>t</eb:mi> </eb:mrow> </eb:msub> </eb:math> and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.