Local Fractional Strong Metric Dimension of Certain Complex Networks

Abstract

Fractional variants of distance-based parameters have application in the fields of sensor networking, robot navigation, and integer programming problems. Complex networks are exceptional networks which exhibit significant topological features and have become quintessential research area in the field of computer science, biology, and mathematics. Owing to the possibility that many real-world systems can be intelligently modeled and represented as complex networks to examine, administer and comprehend the useful information from these real-world networks. In this paper, local fractional strong metric dimension of certain complex networks is computed. Building blocks of complex networks are considered as the symmetric networks such as cyclic networks <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msub> <a:mrow> <a:mi>C</a:mi> </a:mrow> <a:mrow> <a:mi>n</a:mi> </a:mrow> </a:msub> </a:math> , circulant networks <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msub> <c:mrow> <c:mi>C</c:mi> </c:mrow> <c:mrow> <c:mi>n</c:mi> </c:mrow> </c:msub> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mn>1,2</c:mn> </c:mrow> </c:mfenced> </c:math> , mobious ladder networks <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"> <h:msub> <h:mrow> <h:mi>M</h:mi> </h:mrow> <h:mrow> <h:mn>2</h:mn> <h:mi>n</h:mi> </h:mrow> </h:msub> </h:math> , and generalized prism networks <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:msubsup> <j:mrow> <j:mi>G</j:mi> </j:mrow> <j:mrow> <j:mi>m</j:mi> </j:mrow> <j:mrow> <j:mi>n</j:mi> </j:mrow> </j:msubsup> </j:math> . In this regard, it is shown that LSFMD of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:msub> <l:mrow> <l:mi>C</l:mi> </l:mrow> <l:mrow> <l:mi>n</l:mi> </l:mrow> </l:msub> <l:mfenced open="(" close=")" separators="|"> <l:mrow> <l:mi>n</l:mi> <l:mo>≥</l:mo> <l:mn>3</l:mn> </l:mrow> </l:mfenced> </l:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M6"> <q:msubsup> <q:mrow> <q:mi>G</q:mi> </q:mrow> <q:mrow> <q:mi>m</q:mi> </q:mrow> <q:mrow> <q:mi>n</q:mi> </q:mrow> </q:msubsup> <q:mfenced open="(" close=")" separators="|"> <q:mrow> <q:mi>n</q:mi> <q:mo>≥</q:mo> <q:mn>6</q:mn> </q:mrow> </q:mfenced> </q:math> is 1 when <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M7"> <v:mi>n</v:mi> </v:math> is even and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M8"> <x:mi>n</x:mi> <x:mo>/</x:mo> <x:mi>n</x:mi> <x:mo>−</x:mo> <x:mn>1</x:mn> </x:math> when <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M9"> <z:mi>n</z:mi> </z:math> is odd, whereas LSFMD of <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M10"> <bb:msub> <bb:mrow> <bb:mi>M</bb:mi> </bb:mrow> <bb:mrow> <bb:mn>2</bb:mn> <bb:mi>n</bb:mi> </bb:mrow> </bb:msub> </bb:math> is 1 when <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" id="M11"> <db:mi>n</db:mi> </db:math> is odd and <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" id="M12"> <fb:mi>n</fb:mi> <fb:mo>/</fb:mo> <fb:mi>n</fb:mi> <fb:mo>−</fb:mo> <fb:mn>1</fb:mn> </fb:math> when <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M13"> <hb:mi>n</hb:mi> </hb:math> is even. Also, LSFMD of <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" id="M14"> <jb:msub> <jb:mrow> <jb:mi>C</jb:mi> </jb:mrow> <jb:mrow> <jb:mi>n</jb:mi> </jb:mrow> </jb:msub> <jb:mfenced open="(" close=")" separators="|"> <jb:mrow> <jb:mn>1,2</jb:mn> </jb:mrow> </jb:mfenced> </jb:math> is <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M15"> <ob:mi>n</ob:mi> <ob:mo>/</ob:mo> <ob:mn>2</ob:mn> <ob:mrow> <ob:mfenced open="(" close=")" separators="|"> <ob:mrow> <ob:mo>⌈</ob:mo> <ob:mrow> <ob:mrow> <ob:mi>m</ob:mi> <ob:mo>+</ob:mo> <ob:mn>1</ob:mn> </ob:mrow> <ob:mo>/</ob:mo> <ob:mn>2</ob:mn> </ob:mrow> <ob:mo>⌉</ob:mo> </ob:mrow> </ob:mfenced> </ob:mrow> </ob:math> where <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M16"> <tb:mi>n</tb:mi> <tb:mo>≥