On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

Given a finite commutative unital ring <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="script">S</a:mi> </a:math> having some non-zero elements <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:mi>x</d:mi> <d:mo>,</d:mo> <d:mtext> </d:mtext> <d:mi>y</d:mi> </d:math> such that <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M3"> <f:mi>x</f:mi> <f:mo>.</f:mo> <f:mi>y</f:mi> <f:mo>=</f:mo> <f:mn>0</f:mn> </f:math> , the elements of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M4"> <h:mi mathvariant="script">S</h:mi> </h:math> that possess such property are called the zero divisors, denoted by <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M5"> <k:mi mathvariant="script">Z</k:mi> <k:mfenced open="(" close=")" separators="|"> <k:mrow> <k:mi>S</k:mi> </k:mrow> </k:mfenced> </k:math> . We can associate a graph to <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M6"> <q:mi mathvariant="script">S</q:mi> </q:math> with the help of zero-divisor set <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M7"> <t:mi mathvariant="script">Z</t:mi> <t:mfenced open="(" close=")" separators="|"> <t:mrow> <t:mi>S</t:mi> </t:mrow> </t:mfenced> </t:math> , denoted by <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M8"> <z:mi>ζ</z:mi> <z:mfenced open="(" close=")" separators="|"> <z:mrow> <z:mi>S</z:mi> </z:mrow> </z:mfenced> </z:math> (called the zero-divisor graph), to study the algebraic properties of the ring <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M9"> <eb:mi mathvariant="script">S</eb:mi> </eb:math> . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M10"> <hb:mi mathvariant="script">S</hb:mi> </hb:math> . To do so, we will discuss the zero-divisor graphs for the ring of integers <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M11"> <kb:msub> <kb:mrow> <kb:mi>ℤ</kb:mi> </kb:mrow> <kb:mrow> <kb:mi>m</kb:mi> </kb:mrow> </kb:msub> </kb:math> modulo <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M12"> <mb:mi>m</mb:mi> </mb:math> , some quotient polynomial rings, and the ring of Gaussian integers <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M13"> <ob:msub> <ob:mrow> <ob:mi>ℤ</ob:mi> </ob:mrow> <ob:mrow> <ob:mi>m</ob:mi> </ob:mrow> </ob:msub> <ob:mfenced open="[" close="]" separators="|"> <ob:mrow> <ob:mi>i</ob:mi> </ob:mrow> </ob:mfenced> </ob:math> modulo <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M14"> <tb:mi>m</tb:mi> </tb:math> . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M15"> <vb:mi>ζ</vb:mi> <vb:mfenced open="(" close=")" separators="|"> <vb:mrow> <vb:mi>S</vb:mi> </vb:mrow> </vb:mfenced> </vb:math> . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.