AUTHOR=Wei Yinhu , Luo Ricai 

TITLE=The wiener index of the zero-divisor graph for a new class of residue class rings

JOURNAL=Frontiers in Chemistry

VOLUME=10

YEAR=2022

URL=https://www.frontiersin.org/journals/chemistry/articles/10.3389/fchem.2022.985001

DOI=10.3389/fchem.2022.985001

ISSN=2296-2646

ABSTRACT=<p>The zero-divisor graph of a commutative ring R, denoted by Γ(<italic>R</italic>), is a graph whose two distinct vertices <italic>x</italic> and <italic>y</italic> are joined by an edge if and only if <italic>xy</italic> = 0 or <italic>yx</italic> = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo <italic>p</italic><sup><italic>s</italic></sup><italic>q</italic><sup><italic>t</italic></sup> for all distinct primes <italic>p</italic>, <italic>q</italic> and <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">N</mml:mi></mml:math></inline-formula>. We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(<italic>R</italic>), and a formula for the Wiener index of associated compressed zero-divisor graph Γ<sub><italic>E</italic></sub>(<italic>R</italic>) are derived for <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>.</p>