In our research, we are studying the relationship between the energy and Zagreb indices of a graph. We have proven several results, including: (i) tight lower and upper bounds for the energy of graphs based on their order, size, minimum degree, maximum degree, minimum eigenvalue, Zagreb indices, positive inertia, and negative inertia. We have also characterized the graphs that achieve equalities. (ii) Tight lower and upper bounds for maximum eigenvalue of graphs in terms of their order, size, minimum degree, maximum degree, and Zagreb indices. We have also characterized the graphs that achieve equalities. After conducting observations and computational calculations, we have formulated the conjecture that for a non-singular graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2376_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with order p, size m, and the first Zagreb index <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2376_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$M_1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the following should hold: <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2376_Article_IEq3.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}(\Omega )\ge \frac{M_1(\Omega )}{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mfrac> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>m</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2376_Article_IEq4.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {E}(\Omega )\ge \frac{M_1(\Omega )}{2m}+\frac{2m}{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mfrac> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>p</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, with both equalities holding iff <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2376_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \cong \,K_p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>≅</mo> <mspace width="0.166667em" /> <msub> <mi>K</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. It has been demonstrated that proving the first inequality will validate the second inequality.

On the connection between energy and Zagreb indices of graphs

期刊论文
发表日期：

2025年1月14日

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Kinkar Chandra Das,
Ali Ghalavand

摘要

In our research, we are studying the relationship between the energy and Zagreb indices of a graph. We have proven several results, including: (i) tight lower and upper bounds for the energy of graphs based on their order, size, minimum degree, maximum degree, minimum eigenvalue, Zagreb indices, positive inertia, and negative inertia. We have also characterized the graphs that achieve equalities. (ii) Tight lower and upper bounds for maximum eigenvalue of graphs in terms of their order, size, minimum degree, maximum degree, and Zagreb indices. We have also characterized the graphs that achieve equalities. After conducting observations and computational calculations, we have formulated the conjecture that for a non-singular graph $\Omega $ $Ω$ with order p, size m, and the first Zagreb index $M_1(\Omega )$ $M_{1} (Ω)$ , the following should hold: $\mathcal {E}(\Omega )\ge \frac{M_1(\Omega )}{m}$ $E (Ω) \geq \frac{M_{1} (Ω)}{m}$ and $\mathcal {E}(\Omega )\ge \frac{M_1(\Omega )}{2m}+\frac{2m}{p}$ $E (Ω) \geq \frac{M_{1} (Ω)}{2 m} + \frac{2 m}{p}$ , with both equalities holding iff $\Omega \cong \,K_p$ $Ω ≅ K_{p}$ . It has been demonstrated that proving the first inequality will validate the second inequality.