Energy and Randic' energy of special graphs

TL;DR

This paper calculates the Randic' energy for various special graph constructions and explores their properties, enabling the creation of infinite families of graphs with identical or specific energy characteristics.

Contribution

It introduces formulas for Randic' energy of m-splitting, m-shadow, and m-duplicate graphs, and discusses methods to generate infinite equienergetic graph families.

Findings

Derived Randic' energy formulas for special graph types

Constructed infinite sequences of equienergetic graphs

Identified methods to obtain integral graphs and graphs with specific invariants

Abstract

In this paper, we determine the Randic' energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randic' energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny's constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randic' equienergetic graphs and also, infinite families of integral graphs.