SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

作者：Milovanovic I; Matejic M; Glogic E; Milovanovic E

来源：Bulletin of the Australian Mathematical Society, 2018, 97(1): 1-10.

DOI：10.1017/S0004972717000831

<jats:p>Let <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline1" xlink:type="simple" />
<jats:tex-math>$G$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> be a simple connected graph with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline2" xlink:type="simple" />
<jats:tex-math>$n$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> vertices and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline3" xlink:type="simple" />
<jats:tex-math>$m$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> edges and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline4" xlink:type="simple" />
<jats:tex-math>$d_{1}\geq d_{2}\geq \cdots \geq d_{n}&amp;gt;0$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> its sequence of vertex degrees. If <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline5" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}&amp;gt;\unicode[STIX]{x1D707}_{n}=0$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> are the Laplacian eigenvalues of <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline6" xlink:type="simple" />
<jats:tex-math>$G$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, then the Kirchhoff index of <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline7" xlink:type="simple" />
<jats:tex-math>$G$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline8" xlink:type="simple" />
<jats:tex-math>$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. We prove some new lower bounds for <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline9" xlink:type="simple" />
<jats:tex-math>$\mathit{Kf}(G)$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> in terms of some of the parameters <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline10" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D6E5}=d_{1}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline11" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D6E5}_{2}=d_{2}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline12" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D6E5}_{3}=d_{3}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline13" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D6FF}=d_{n}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline14" xlink:type="simple" />
<jats:tex-math>$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and the topological index <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0004972717000831_inline15" xlink:type="simple" />
<jats:tex-math>$\mathit{NK}=\prod _{i=1}^{n}d_{i}$</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.</jats:p>

• 出版日期2018-2