Given a connected graph G with the vertex set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation>, the eccentric distance sum (EDS) of G is defined as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sum _{x\in V_G} {\varepsilon }_{\!_G}(x)D_{\!_G}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>x</mi> <mo>∈</mo> <msub> <mi>V</mi> <mi>G</mi> </msub> </mrow> </msub> <mmultiscripts> <mi>ε</mi> <mmultiscripts> <mspace width="-0.166667em" /> <mi>G</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mmultiscripts> <mi>D</mi> <mmultiscripts> <mspace width="-0.166667em" /> <mi>G</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\varepsilon }_{\!_G}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>ε</mi> <mmultiscripts> <mspace width="-0.166667em" /> <mi>G</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the eccentricity of the vertex x and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D_{\!_G}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>D</mi> <mmultiscripts> <mspace width="-0.166667em" /> <mi>G</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the sum of distances from x to all other vertices of G. In this paper, we characterize the graphs with the maximum EDS and the minimum EDS, respectively, among the class of all trees having a fixed number of pendant paths.

On the eccentric distance sums of trees with fixed number of pendant paths

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发表日期：

2025年12月22日

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R. Azhagendran,
T. Divyadevi,
I. Jeyaraman

摘要

Given a connected graph G with the vertex set $V_G$ $V_{G}$ , the eccentric distance sum (EDS) of G is defined as $\sum _{x\in V_G} {\varepsilon }_{\!_G}(x)D_{\!_G}(x)$ $\sum_{x \in V_{G}} ε_{_{G}} (x) D_{_{G}} (x)$ , where ${\varepsilon }_{\!_G}(x)$ $ε_{_{G}} (x)$ is the eccentricity of the vertex x and $D_{\!_G}(x)$ $D_{_{G}} (x)$ is the sum of distances from x to all other vertices of G. In this paper, we characterize the graphs with the maximum EDS and the minimum EDS, respectively, among the class of all trees having a fixed number of pendant paths.