Some Inequalities between General Randić-Type Graph Invariants

The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>G</a:mi></a:math> with vertex set <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>V</c:mi></c:math> as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:msub><e:mrow><e:mi>R</e:mi></e:mrow><e:mrow><e:mi>α</e:mi></e:mrow></e:msub><e:mfenced open="(" close=")" separators="|"><e:mrow><e:mi>G</e:mi></e:mrow></e:mfenced><e:mo>=</e:mo><e:msub><e:mrow><e:mstyle displaystyle="true"><e:mo stretchy="false">∑</e:mo></e:mstyle></e:mrow><e:mrow><e:msub><e:mrow><e:mi>υ</e:mi></e:mrow><e:mrow><e:mi>i</e:mi></e:mrow></e:msub><e:mo>∼</e:mo><e:msub><e:mrow><e:mi>υ</e:mi></e:mrow><e:mrow><e:mi>j</e:mi></e:mrow></e:msub></e:mrow></e:msub><e:msup><e:mrow><e:mfenced open="(" close=")" separators="|"><e:mrow><e:msub><e:mrow><e:mi>d</e:mi></e:mrow><e:mrow><e:mi>i</e:mi></e:mrow></e:msub><e:msub><e:mrow><e:mi>d</e:mi></e:mrow><e:mrow><e:mi>j</e:mi></e:mrow></e:msub></e:mrow></e:mfenced></e:mrow><e:mrow><e:mi>α</e:mi></e:mrow></e:msup></e:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M4"><o:msub><o:mrow><o:mi>Q</o:mi></o:mrow><o:mrow><o:mi>α</o:mi></o:mrow></o:msub><o:mfenced open="(" close=")" separators="|"><o:mrow><o:mi>G</o:mi></o:mrow></o:mfenced><o:mo>=</o:mo><o:msub><o:mrow><o:mstyle displaystyle="true"><o:mo stretchy="false">∑</o:mo></o:mstyle></o:mrow><o:mrow><o:msub><o:mrow><o:mi>v</o:mi></o:mrow><o:mrow><o:mi>i</o:mi></o:mrow></o:msub><o:mo>∈</o:mo><o:mi>V</o:mi></o:mrow></o:msub><o:msubsup><o:mrow><o:mi>d</o:mi></o:mrow><o:mrow><o:mi>i</o:mi></o:mrow><o:mrow><o:mi>α</o:mi></o:mrow></o:msubsup></o:math> , respectively, where <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M5"><v:mi>α</v:mi></v:math> is an arbitrary real number, <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M6"><x:msub><x:mrow><x:mi>d</x:mi></x:mrow><x:mrow><x:mi>i</x:mi></x:mrow></x:msub></x:math> denotes the degree of a vertex <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M7"><z:msub><z:mrow><z:mi>υ</z:mi></z:mrow><z:mrow><z:mi>i</z:mi></z:mrow></z:msub></z:math> , and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M8"><bb:msub><bb:mrow><bb:mi>υ</bb:mi></bb:mrow><bb:mrow><bb:mi>i</bb:mi></bb:mrow></bb:msub><bb:mo>∼</bb:mo><bb:msub><bb:mrow><bb:mi>υ</bb:mi></bb:mrow><bb:mrow><bb:mi>j</bb:mi></bb:mrow></bb:msub></bb:math> represents the adjacency of vertices <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" id="M9"><db:msub><db:mrow><db:mi>υ</db:mi></db:mrow><db:mrow><db:mi>i</db:mi></db:mrow></db:msub></db:math> and <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" id="M10"><fb:msub><fb:mrow><fb:mi>υ</fb:mi></fb:mrow><fb:mrow><fb:mi>j</fb:mi></fb:mrow></fb:msub></fb:math> in <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M11"><hb:mi>G</hb:mi></hb:math> . Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" id="M12"><jb:msub><jb:mrow><jb:mi>R</jb:mi></jb:mrow><jb:mrow><jb:mi>α</jb:mi></jb:mrow></jb:msub></jb:math> and <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M13"><lb:msub><lb:mrow><lb:mi>Q</lb:mi></lb:mrow><lb:mrow><lb:mi>α</lb:mi></lb:mrow></lb:msub></lb:math> have been derived so far. In this paper, we establish explicit inequality relationships between <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M14"><nb:msub><nb:mrow><nb:mi>R</nb:mi></nb:mrow><nb:mrow><nb:mi>α</nb:mi></nb:mrow></nb:msub></nb:math> and <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M15"><pb:msub><pb:mrow><pb:mi>Q</pb:mi></pb:mrow><pb:mrow><pb:mi>α</pb:mi></pb:mrow></pb:msub></pb:math> . Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M16"><rb:mi>α</rb:mi></rb:math> .