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<p>Centrality measures are used to quantify the influence or importance of vertices within a graph. The stress of an internal vertex <i>u</i> in a connected graph <i>G</i> is a centrality measure defined as the number of shortest paths passing through a vertex <i>u</i> in a graph <i>G</i>. In this paper, we introduce a new vertex-based invariant, complementing to stress, called relief of a vertex <i>u</i> in a connected graph <i>G</i> denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Re_{G}(u)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and is defined as the number of shortest paths not passing through a vertex <i>u</i> in a graph <i>G</i>. The sum of relief of all vertices of a connected graph <i>G</i> is called relief of <i>G</i> denoted by <i>Re</i>(<i>G</i>). We obtain the relief of a vertex in a tree and hence obtain the relief of trees. We construct a tree where relief and stress of a vertex is the same. Further linear regression analysis of the relief with the physico-chemical properties of alkanes is carried out. The linear model, based on the relief shows good correlation with the physico-chemical properties of alkanes.</p>

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Relief of Trees and Its Application

  • 期刊论文

  • 发表日期：

    2026年1月20日

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  • Harishchandra S. Ramane,
  • Mallappa P. Mellikeri

摘要

Centrality measures are used to quantify the influence or importance of vertices within a graph. The stress of an internal vertex u in a connected graph G is a centrality measure defined as the number of shortest paths passing through a vertex u in a graph G. In this paper, we introduce a new vertex-based invariant, complementing to stress, called relief of a vertex u in a connected graph G denoted by \(Re_{G}(u)\) R e G ( u ) and is defined as the number of shortest paths not passing through a vertex u in a graph G. The sum of relief of all vertices of a connected graph G is called relief of G denoted by Re(G). We obtain the relief of a vertex in a tree and hence obtain the relief of trees. We construct a tree where relief and stress of a vertex is the same. Further linear regression analysis of the relief with the physico-chemical properties of alkanes is carried out. The linear model, based on the relief shows good correlation with the physico-chemical properties of alkanes.