<p>The total graph of a finite commutative ring with unity, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T(\Gamma (R))\)</EquationSource> </InlineEquation>, consists of vertices representing the elements of <i>R</i>. Two distinct vertices <i>x</i> and <i>y</i> are connected if their sum <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x+y\)</EquationSource> </InlineEquation> belongs to the set <i>Z</i>(<i>R</i>), which represents the set of zero-divisors of <i>R</i>. This paper provides the necessary condition for the existence of the metric dimension of total graph of a finite commutative ring. A bound for the metric dimension has been obtained and with this context rings have been characterized for which the metric dimension of total graph attains a sharp bound. Additionally, the relationship between the metric dimension of total graph and other graph invariants have been developed.</p>

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Metric Dimension of Total Graph of a Finite Commutative Ring

  • Pranjali,
  • Amit Kumar,
  • Rakshita Sharma

摘要

The total graph of a finite commutative ring with unity, denoted by \(T(\Gamma (R))\) , consists of vertices representing the elements of R. Two distinct vertices x and y are connected if their sum \(x+y\) belongs to the set Z(R), which represents the set of zero-divisors of R. This paper provides the necessary condition for the existence of the metric dimension of total graph of a finite commutative ring. A bound for the metric dimension has been obtained and with this context rings have been characterized for which the metric dimension of total graph attains a sharp bound. Additionally, the relationship between the metric dimension of total graph and other graph invariants have been developed.