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On Divided-Type Connectivity of Graphs.

Qiao Zhou1, Xiaomin Wang2, Bing Yao3,4

  • 1College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, China.

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|January 21, 2023
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This study introduces novel graph operations and a new metric, divided connectivity, to analyze graph structures and reliability. These methods offer new insights into graph vulnerability and fault tolerance for interconnection networks.

Keywords:
Euler graphcoincident operationdivided connectivitydivided operation

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Area of Science:

  • Graph Theory
  • Network Analysis
  • Computer Science

Background:

  • Graph connectivity is crucial for network reliability and fault tolerance.
  • Modern interconnection graphs require robust metrics for vulnerability assessment.

Purpose of the Study:

  • Introduce novel vertex- and edge-divided operations for graph manipulation.
  • Define and explore a new metric: divided connectivity.
  • Establish an equivalence between traditional and divided connectivity.

Main Methods:

  • Defined vertex-divided and edge-divided operations and their inverses.
  • Introduced the concept of divided connectivity.
  • Investigated graph structures using vertex-divided connectivity.
  • Applied divided operations to identify conditions for Euler's graphs.

Main Results:

  • Presented methods for splitting graph vertices.
  • Established an equivalence relationship between traditional and divided connectivity.
  • Explored graph structures based on vertex-divided connectivity.
  • Provided conditions for Euler's graphs using divided operations.

Conclusions:

  • The novel divided operations and divided connectivity offer new perspectives on graph analysis.
  • These concepts enhance understanding of graph vulnerability, reliability, and fault tolerance.
  • Further research can explore additional applications and properties of divided connectivity.