The degree variance and geometric degree variance of random trees

Document Type : Research Paper

Author

Department of Statistics, Imam Khomeini International University, Qazvin, Iran

Abstract

The degree variance and the geometric mean of the degrees of the vertices for ‎graph $G‎$ are defined ‎as ‎‎‎$\text{Var}(G)=‎\frac{1}{n}\sum_{i=1}^{n}\Big(d(v_i)-‎\frac{2m}{n}‎ \Big)^2‎‎$‎‎ and ‎‎$‎\text{GM}(G)=\Big(\prod_{i=1}^{n}‎d(v_i)\Big)‎^‎\frac{1}{n}‎,‎$‎ respectively, ‎w‎here $n$, $m$ and $d(v)$ represent the number of vertices, edges and degree of vertex $v$. ‎Also,‎ the geometric degree variance of graph ‎$‎‎G$ ‎defined as ‎‎‎$‎‎\text{GVAR}(G)=‎\frac{1}{n}‎\sum_{i=1}^{n}(d(v_i)-\text{GM}(G))^2‎‎$‎‎. ‎‎‎ We determine the two first moment of degree variance in (uniform) random tree‏‎s.‎ We also show a convergence in probability associated with this quantity. ‎Finally, we present bounds for the ‎expected ‎value‎ of geometric mean of the degrees ‎and‎ geometric degree variance.

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References

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History

  • Receive Date: 12 October 2025
  • Revise Date: 17 December 2025
  • Accept Date: 02 February 2026

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