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Randić energy of digraphs.

Roberto Cruz1, Juan Monsalve1, Juan Rada1

  • 1Instituto de Matemáticas, Universidad de Antioquia, Medellín, Colombia.

Heliyon
|December 2, 2022
PubMed
Summary

This study introduces a new definition for graph energy using VDB topological indices and explores its properties, particularly for the Randić index in digraphs. New bounds for Randić energy are derived, offering insights into graph structures.

Area of Science:

  • Graph Theory
  • Spectral Graph Theory
  • Network Analysis

Background:

  • Topological indices quantify graph properties.
  • Graph energy is a measure of molecular stability and has applications in chemistry and physics.
  • Existing graph energy definitions often rely on adjacency or Laplacian matrices.

Purpose of the Study:

  • To define and investigate a new concept of graph energy for directed graphs (digraphs) using VDB topological indices.
  • To analyze the properties of this new energy measure, especially when the VDB index is the Randić index.
  • To derive novel upper and lower bounds for Randić energy in digraphs and graphs.

Main Methods:

  • Definition of VDB topological index and associated energy for digraphs.
  • Analysis of the spectral norm and rank of the generalized adjacency matrix for the Randić index.
Keywords:
DigraphsRandić energyRandić indexVDB topological index

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  • Derivation of new energy bounds using matrix properties and graph structure.

    • Main Results:

    • For the Randić index, the spectral norm of the generalized adjacency matrix is 1, and its rank equals the adjacency matrix rank.
    • These properties do not generally hold for other VDB topological indices.
    • New upper and lower bounds for Randić energy in digraphs are established, with some generalizing existing graph results.

    Conclusions:

    • The Randić matrix possesses unique properties beneficial for energy analysis in digraphs.
    • A new upper bound for Randić energy in graphs is derived, linked to the graph's rank.
    • The findings contribute to a deeper understanding of graph energy and its relationship to graph structure.