Edge Metric Dimension of Some Classes of Toeplitz Networks

Abstract

Toeplitz networks are used as interconnection networks due to their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The edge metric dimension of a network is recently introduced, and its applications can be seen in several areas including robot navigation, intelligent systems, network designing, and image processing. For a vertex <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>s</a:mi> </a:math> and an edge <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>g</c:mi> <c:mo>=</c:mo> <c:msub> <c:mrow> <c:mi>s</c:mi> </c:mrow> <c:mrow> <c:mn>1</c:mn> </c:mrow> </c:msub> <c:msub> <c:mrow> <c:mi>s</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:math> of a connected graph <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mi>G</e:mi> </e:math> , the minimum number from distances of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"> <g:mi>s</g:mi> </g:math> with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M5"> <i:msub> <i:mrow> <i:mi>s</i:mi> </i:mrow> <i:mrow> <i:mn>1</i:mn> </i:mrow> </i:msub> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M6"> <k:msub> <k:mrow> <k:mi>s</k:mi> </k:mrow> <k:mrow> <k:mn>2</k:mn> </k:mrow> </k:msub> </k:math> is called the distance between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M7"> <m:mi>s</m:mi> </m:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M8"> <o:mi>g</o:mi> </o:math> . If for every two distinct edges <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M9"> <q:msub> <q:mrow> <q:mi>s</q:mi> </q:mrow> <q:mrow> <q:mn>1</q:mn> </q:mrow> </q:msub> <q:mo>,</q:mo> <q:msub> <q:mrow> <q:mi>s</q:mi> </q:mrow> <q:mrow> <q:mn>2</q:mn> </q:mrow> </q:msub> <q:mo>∈</q:mo> <q:mi>E</q:mi> <q:mfenced open="(" close=")" separators="|"> <q:mrow> <q:mi>G</q:mi> </q:mrow> </q:mfenced> </q:math> , there always exists <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M10"> <v:msub> <v:mrow> <v:mi>w</v:mi> </v:mrow> <v:mrow> <v:mn>1</v:mn> </v:mrow> </v:msub> <v:mi>ɛ</v:mi> <v:msub> <v:mrow> <v:mi>W</v:mi> </v:mrow> <v:mrow> <v:mi>E</v:mi> </v:mrow> </v:msub> <v:mo>⊆</v:mo> <v:mi>V</v:mi> <v:mfenced open="(" close=")" separators="|"> <v:mrow> <v:mi>G</v:mi> </v:mrow> </v:mfenced> </v:math> , such that <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M11"> <ab:mi>d</ab:mi> <ab:mfenced open="(" close=")" separators="|"> <ab:mrow> <ab:msub> <ab:mrow> <ab:mi>s</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>1</ab:mn> </ab:mrow> </ab:msub> <ab:mo>,</ab:mo> <ab:msub> <ab:mrow> <ab:mi>w</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>1</ab:mn> </ab:mrow> </ab:msub> </ab:mrow> </ab:mfenced> <ab:mo>≠</ab:mo> <ab:mi>d</ab:mi> <ab:mfenced open="(" close=")" separators="|"> <ab:mrow> <ab:msub> <ab:mrow> <ab:mi>s</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>2</ab:mn> </ab:mrow> </ab:msub> <ab:mo>,</ab:mo> <ab:msub> <ab:mrow> <ab:mi>w</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>1</ab:mn> </ab:mrow> </ab:msub> </ab:mrow> </ab:mfenced> </ab:math> ; then, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M12"> <ib:msub> <ib:mrow> <ib:mi>W</ib:mi> </ib:mrow> <ib:mrow> <ib:mi>E</ib:mi> </ib:mrow> </ib:msub> </ib:math> is named as an edge metric generator. The minimum number of vertices in <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M13"> <kb:msub> <kb:mrow> <kb:mi>W</kb:mi> </kb:mrow> <kb:mrow> <kb:mi>E</kb:mi> </kb:mrow> </kb:msub> </kb:math> is known as the edge metric dimension of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M14"> <mb:mi>G</mb:mi> </mb:math> . In this study, we consider four families of Toeplitz networks <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M15"> <ob:msub> <ob:mrow> <ob:mi>T</ob:mi> </ob:mrow> <ob:mrow> <ob:mi>n</ob:mi> </ob:mrow> </ob:msub> <ob:mfenced open="(" close=")" separators="|"> <ob:mrow> <ob:mn>1,2</ob:mn> </ob:mrow> </ob:mfenced> </ob:math>