Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach

Abstract

The algebraic polynomial plays a significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. The topological index is utilized as a significant tool in the study of the quantitative structure activity relationship (QSAR) and quantitative structures property relationship (QSPR) which correlate a molecular structure to its different properties and activities. Graphs containing finite commutative rings have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this article, the topological indices of the total graph <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>T</a:mi> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:msub> <a:mrow> <a:mi>ℤ</a:mi> </a:mrow> <a:mrow> <a:mi>n</a:mi> </a:mrow> </a:msub> </a:mrow> </a:mfenced> </a:math> <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mi>n</f:mi> <f:mo>∈</f:mo> <f:msup> <f:mrow> <f:mi>ℤ</f:mi> </f:mrow> <f:mrow> <f:mo>+</f:mo> </f:mrow> </f:msup> </f:mrow> </f:mfenced> </f:math> , the zero divisor graph <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"> <k:mi mathvariant="normal">Γ</k:mi> <k:mfenced open="(" close=")" separators="|"> <k:mrow> <k:msub> <k:mrow> <k:mi>ℤ</k:mi> </k:mrow> <k:mrow> <k:msup> <k:mrow> <k:mi>r</k:mi> </k:mrow> <k:mrow> <k:mi>n</k:mi> </k:mrow> </k:msup> </k:mrow> </k:msub> </k:mrow> </k:mfenced> </k:math> ( <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M4"> <q:mi>r</q:mi> </q:math> is prime, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M5"> <s:mi>n</s:mi> <s:mo>∈</s:mo> <s:msup> <s:mrow> <s:mi>ℤ</s:mi> </s:mrow> <s:mrow> <s:mo>+</s:mo> </s:mrow> </s:msup> </s:math> ), and the zero divisor graph <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M6"> <u:mi mathvariant="normal">Γ</u:mi> <u:mfenced open="(" close=")" separators="|"> <u:mrow> <u:msub> <u:mrow> <u:mi>ℤ</u:mi> </u:mrow> <u:mrow> <u:mi>r</u:mi> </u:mrow> </u:msub> <u:mo>×</u:mo> <u:msub> <u:mrow> <u:mi>ℤ</u:mi> </u:mrow> <u:mrow> <u:mi>s</u:mi> </u:mrow> </u:msub> <u:mo>×</u:mo> <u:msub> <u:mrow> <u:mi>ℤ</u:mi> </u:mrow> <u:mrow> <u:mi>t</u:mi> </u:mrow> </u:msub> </u:mrow> </u:mfenced> </u:math> ( <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M7"> <ab:mi>r</ab:mi> <ab:mo>,</ab:mo> <ab:mi>s</ab:mi> <ab:mo>,</ab:mo> <ab:mi>t</ab:mi> </ab:math> are primes) are computed using some algebraic polynomials.