First General Zagreb Index of Generalized F-sum Graphs

Abstract

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>G</a:mi> </a:math> can be defined as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msup> <c:mrow> <c:mi>M</c:mi> </c:mrow> <c:mrow> <c:mi>γ</c:mi> </c:mrow> </c:msup> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>G</c:mi> </c:mrow> </c:mfenced> <c:mo>=</c:mo> <c:mstyle displaystyle="true"> <c:msub> <c:mrow> <c:mo stretchy="false">∑</c:mo> </c:mrow> <c:mrow> <c:mi>u</c:mi> <c:mi>v</c:mi> <c:mo>∈</c:mo> <c:mi>E</c:mi> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>G</c:mi> </c:mrow> </c:mfenced> </c:mrow> </c:msub> <c:mfenced open="[" close="]" separators="|"> <c:mrow> <c:msubsup> <c:mi>d</c:mi> <c:mi>G</c:mi> <c:mrow> <c:mi>γ</c:mi> <c:mo>−</c:mo> <c:mn>1</c:mn> </c:mrow> </c:msubsup> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>u</c:mi> </c:mrow> </c:mfenced> <c:mo>+</c:mo> <c:msubsup> <c:mi>d</c:mi> <c:mi>G</c:mi> <c:mrow> <c:mi>γ</c:mi> <c:mo>−</c:mo> <c:mn>1</c:mn> </c:mrow> </c:msubsup> <c:mfenced open="(" close=")" separators="|"> <c:mrow> <c:mi>v</c:mi> </c:mrow> </c:mfenced> </c:mrow> </c:mfenced> </c:mstyle> </c:math> , where <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M3"> <v:mi>γ</v:mi> </v:math> is a real number. As <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M4"> <x:msup> <x:mrow> <x:mi>M</x:mi> </x:mrow> <x:mrow> <x:mi>γ</x:mi> </x:mrow> </x:msup> <x:mfenced open="(" close=")" separators="|"> <x:mrow> <x:mi>G</x:mi> </x:mrow> </x:mfenced> </x:math> is equal to the order and size of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M5"> <cb:mi>G</cb:mi> </cb:math> when <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M6"> <eb:mi>γ</eb:mi> <eb:mo>=</eb:mo> <eb:mn>0</eb:mn> </eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M7"> <gb:mi>γ</gb:mi> <gb:mo>=</gb:mo> <gb:mn>1</gb:mn> </gb:math> , respectively, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M8"> <ib:mi>γ</ib:mi> </ib:math> is usually assumed to be different from 0 to 1. In this paper, for every integer <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M9"> <kb:mi>γ</kb:mi> <kb:mo>≥</kb:mo> <kb:mn>2</kb:mn> </kb:math> , the FGZ index <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M10"> <mb:msup> <mb:mrow> <mb:mi>M</mb:mi> </mb:mrow> <mb:mrow> <mb:mi>γ</mb:mi> </mb:mrow> </mb:msup> </mb:math> is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).

Topics & Concepts