Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

Let <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="script">A</a:mi> </a:math> be a commutative ring with unity and let set of all zero divisors of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:mi mathvariant="script">A</d:mi> </d:math> be denoted by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M3"> <g:mi mathvariant="script">Z</g:mi> <g:mfenced open="(" close=")" separators="|"> <g:mrow> <g:mi mathvariant="script">A</g:mi> </g:mrow> </g:mfenced> </g:math> . An ideal <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M4"> <n:mi mathvariant="normal">ℐ</n:mi> </n:math> of the ring <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M5"> <q:mi mathvariant="script">A</q:mi> </q:math> is said to be essential if it has a nonzero intersection with every nonzero ideal of <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M6"> <t:mi mathvariant="script">A</t:mi> </t:math> . It is denoted by <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M7"> <w:mi mathvariant="normal">ℐ</w:mi> <w:msub> <w:mrow> <w:mo>≤</w:mo> </w:mrow> <w:mrow> <w:mi>e</w:mi> </w:mrow> </w:msub> <w:mi mathvariant="script">A</w:mi> </w:math> . The generalized zero-divisor graph denoted by <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M8"> <ab:msub> <ab:mi mathvariant="normal">Γ</ab:mi> <ab:mrow> <ab:mi>g</ab:mi> </ab:mrow> </ab:msub> <ab:mfenced open="(" close=")" separators="|"> <ab:mrow> <ab:mi mathvariant="script">A</ab:mi> </ab:mrow> </ab:mfenced> </ab:math> is an undirected graph with vertex set <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M9"> <hb:mi mathvariant="script">Z</hb:mi> <hb:msup> <hb:mrow> <hb:mfenced open="(" close=")" separators="|"> <hb:mrow> <hb:mi mathvariant="script">A</hb:mi> </hb:mrow> </hb:mfenced> </hb:mrow> <hb:mi>∗</hb:mi> </hb:msup> </hb:math> (set of all nonzero zero-divisors of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M10"> <ob:mi mathvariant="script">A</ob:mi> </ob:math> ) and two distinct vertices <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M11"> <rb:msub> <rb:mrow> <rb:mi mathvariant="fraktur">x</rb:mi> </rb:mrow> <rb:mrow> <rb:mn>1</rb:mn> </rb:mrow> </rb:msub> </rb:math> and <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" id="M12"> <ub:msub> <ub:mrow> <ub:mi mathvariant="fraktur">x</ub:mi> </ub:mrow> <ub:mrow> <ub:mn>2</ub:mn> </ub:mrow> </ub:msub> </ub:math> are adjacent if and only if <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" id="M13"> <xb:mtext>ann</xb:mtext> <xb:mfenced open="(" close=")" separators="|"> <xb:mrow> <xb:msub> <xb:mrow> <xb:mi mathvariant="fraktur">x</xb:mi> </xb:mrow> <xb:mrow> <xb:mn>1</xb:mn> </xb:mrow> </xb:msub> </xb:mrow> </xb:mfenced> <xb:mo>+</xb:mo> <xb:mtext>ann</xb:mtext> <xb:mfenced open="(" close=")" separators="|"> <xb:mrow> <xb:msub> <xb:mrow> <xb:mi mathvariant="fraktur">x</xb:mi> </xb:mrow> <xb:mrow> <xb:mn>2</xb:mn> </xb:mrow> </xb:msub> </xb:mrow> </xb:mfenced> <xb:msub> <xb:mrow> <xb:mo>≤</xb:mo> </xb:mrow> <xb:mrow> <xb:mi>e</xb:mi> </xb:mrow> </xb:msub> <xb:mi mathvariant="script">A</xb:mi> </xb:math> . In this paper, first we characterized all the finite commutative rings <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" id="M14"> <ic:mi mathvariant="script">A</ic:mi> </ic:math> for which <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" id="M15"> <lc:msub> <lc:mi mathvariant="normal">Γ</lc:mi> <lc:mrow> <lc:mi>g</lc:mi> </lc:mrow> </lc:msub> <lc:mfenced open="(" close=")" separators="|"> <lc:mrow> <lc:mi mathvariant="script">A</lc:mi> </lc:mrow> </lc:mfenced> </lc:math> is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" id="M16"> <sc:mi mathvariant="script">A</sc:mi> </sc:math> for which <vc:math xmlns:vc="http://www.w3.org/1998/Math/MathML" id="M17"> <vc:msub> <vc:mi mathvariant="normal">Γ</vc:mi> <vc:mrow> <vc:mi>g</vc:mi> </vc:mrow> </vc:msub> <vc:mfenced open="(" close=")" separators="|"> <vc:mrow> <vc:mi mathvariant="script">A</vc:mi> </vc:mrow> </vc:mfenced> </vc:math> is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" id="M18"> <cd:msub> <cd:mi mathvariant="normal">Γ</cd:mi> <cd:mrow> <cd:mi>g</cd:mi> </cd:mrow> </cd:msub> <cd:mfenced open="(" close=")" separators="|"> <cd:mrow> <cd:mi mathvariant="script">A</cd:mi> </cd:mrow> </cd:mfenced> </cd:math> .<