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Graph Energies of Generalized and Shadow-Splitting Graphs

Ronak B. Dudhat, Vinodray J. Kaneria, Kalpesh M. Popat

Abstract

We extend the notions of the m-splitting graph Sm(G) and the m-shadow graph Dm(G) to introduce two new graph operations: the (p, q)-generalized splitting graph Sp,q(G) and the (c, k)-shadow-splitting graph Hc,k(G). We derive the adjacency energy of these constructions and as an application, identify several new infinite families of equienergetic and borderenergetic graphs.

Graph Energies of Generalized and Shadow-Splitting Graphs

Abstract

We extend the notions of the m-splitting graph Sm(G) and the m-shadow graph Dm(G) to introduce two new graph operations: the (p, q)-generalized splitting graph Sp,q(G) and the (c, k)-shadow-splitting graph Hc,k(G). We derive the adjacency energy of these constructions and as an application, identify several new infinite families of equienergetic and borderenergetic graphs.

Paper Structure

This paper contains 7 sections, 14 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Energy of the (p,q)-Generalized Splitting Graph
  4. Energy of the (c,k)-Shadow-Splitting Graph
  5. Construction of Equienergetic Graphs
  6. Construction of Borderenergetic Graphs
  7. Conclusion

Key Result

Theorem 3.2

For any $p, q \ge 1$, the energy of the $(p,q)$-generalized splitting graph is $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The $(2,2)$-generalized splitting graph $\mathcal{S}_{2,2}(C_4)$.
  • Figure 2: The $(2,2)$-shadow-splitting graph $\mathcal{H}_{2,2}(C_4)$.

Theorems & Definitions (30)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Corollary 5.1
  • proof
  • Corollary 5.2
  • proof
  • ...and 20 more