This paper presents sufficient conditions for the toughness of a graph in terms of the first Zagreb index, reciprocal degree distance and forgotten index. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1714_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the number of components of a graph G. For <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1714_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$t\geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> we a call graph t-tough if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1714_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">$t\cdot \omega (G-X)\leqslant |X|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>·</mo> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>-</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>⩽</mo> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, for every <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1714_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$X\subseteq V(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1714_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega (G-X)\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>-</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Topological Indices and Toughness of Graphs

期刊论文
发表日期：

2025年8月2日

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Rajkaran Kori,
Abhyendra Prasad

摘要

This paper presents sufficient conditions for the toughness of a graph in terms of the first Zagreb index, reciprocal degree distance and forgotten index. Let $\omega (G)$ $ω (G)$ be the number of components of a graph G. For $t\geqslant 0$ $t ⩾ 0$ we a call graph t-tough if $t\cdot \omega (G-X)\leqslant |X|$ $t \cdot ω (G - X) ⩽ | X |$ , for every $X\subseteq V(G)$ $X \subseteq V (G)$ with $\omega (G-X)\geqslant 2$ $ω (G - X) ⩾ 2$ .