Let D be a digraph with vertex set V(D) and arc set A(D). For a real function f defined on nonnegative real numbers, the vertex-degree function index <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{f}(D)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the digraph D is defined as <Equation ID="Equ9"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_Equ9.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="255" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} H_{f}(D)=\frac{1}{2}\sum _{u\in V(D)}\left[ f(d_{u}^{+}) +f(d_{u}^{-}) \right] , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>H</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munder> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mfenced close="]" open="["> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>d</mi> <mrow> <mi>u</mi> </mrow> <mo>+</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>d</mi> <mrow> <mi>u</mi> </mrow> <mo>-</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$d_u^+$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>d</mi> <mi>u</mi> <mo>+</mo> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$d_u^-$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>d</mi> <mi>u</mi> <mo>-</mo> </msubsup> </math></EquationSource> </InlineEquation> denote the outdegree and the indegree of u, respectively. In this paper we find the extremal values of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$H_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> among orientations of a given graph G, when f is a convex (or concave) real function on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2024_3017_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\left[ 0,+\infty \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="["> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> </mfenced> </math></EquationSource> </InlineEquation>.

Vertex-degree function index on oriented graphs

期刊论文
发表日期：

2025年1月29日

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Sergio Bermudo,
Roberto Cruz,
Juan Rada

摘要

Let D be a digraph with vertex set V(D) and arc set A(D). For a real function f defined on nonnegative real numbers, the vertex-degree function index $H_{f}(D)$ $H_{f} (D)$ of the digraph D is defined as $\begin{aligned} H_{f}(D)=\frac{1}{2}\sum _{u\in V(D)}\left[ f(d_{u}^{+}) +f(d_{u}^{-}) \right] , \end{aligned}$ $\begin{matrix} H_{f} (D) = \frac{1}{2} \sum_{u \in V (D)} [f (d_{u}^{+}) + f (d_{u}^{-})], \end{matrix}$ where $d_u^+$ $d_{u}^{+}$ and $d_u^-$ $d_{u}^{-}$ denote the outdegree and the indegree of u, respectively. In this paper we find the extremal values of $H_{f}$ $H_{f}$ among orientations of a given graph G, when f is a convex (or concave) real function on $\left[ 0,+\infty \right) $ $[0, + \infty)$ .