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Link Residual Closeness of Harary Graphs

Ch. Dangalchev

TL;DR

This work analyzes link residual closeness as a sensitivity measure of network robustness for Harary graphs $H_{k,n}$. By exploiting automorphisms, distances, and known cycle/path closeness results, it derives explicit closed-form formulas for the residual closeness $R(H_{k,n})$ across several Harary configurations, including cases with even and odd connectivity and various $n$. The key contributions are the exact expressions $R(H_{2n},H_{2p,n},H_{3,2n},H_{5,2n},H_{2p+1,2n},H_{3,2n+1},H_{5,2n+1},H_{2m+1,2n+1})$, each expressed as $R= C -$ a linear-plus-fraction term, where the fractional part involves powers of two and floor-valued parameters. These results provide precise, analyzable measures of vulnerability to single-link failures in Harary graphs, with potential impact on designing robust networks and understanding connectivity-resilience tradeoffs in structured graphs.

Abstract

The study of networks characteristics is an important subject in different fields, like math, chemistry, transportation, social network analysis etc. The residual closeness is one of the most sensitive measure of graphs vulnerability. In this article we calculate the link residual closeness of Harary graphs.

Link Residual Closeness of Harary Graphs

TL;DR

This work analyzes link residual closeness as a sensitivity measure of network robustness for Harary graphs . By exploiting automorphisms, distances, and known cycle/path closeness results, it derives explicit closed-form formulas for the residual closeness across several Harary configurations, including cases with even and odd connectivity and various . The key contributions are the exact expressions , each expressed as a linear-plus-fraction term, where the fractional part involves powers of two and floor-valued parameters. These results provide precise, analyzable measures of vulnerability to single-link failures in Harary graphs, with potential impact on designing robust networks and understanding connectivity-resilience tradeoffs in structured graphs.

Abstract

The study of networks characteristics is an important subject in different fields, like math, chemistry, transportation, social network analysis etc. The residual closeness is one of the most sensitive measure of graphs vulnerability. In this article we calculate the link residual closeness of Harary graphs.
Paper Structure (11 sections, 8 theorems, 62 equations)

This paper contains 11 sections, 8 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Residual closeness of $H_{2,n}$
  3. Residual closeness of $H_{2p,n}$
  4. Residual closeness of $H_{3,2n}$
  5. Residual closeness of $H_{5,2n}$
  6. Residual closeness of $H_{2p+1,2n}$
  7. Residual closeness of $H_{3,2n+1}$
  8. Residual closeness of $H_{5,2n+1}$
  9. Residual closeness of $H_{2m+1,2n+1}$
  10. Conclusion
  11. Appendix A. Proof of Formula 1.

Key Result

Theorem 2.1

The residual closeness of Harary graph $H_{2,n}$ is:

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 7.1
  • Theorem 8.1
  • Theorem 9.1