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Exponentiation of Graphs

Toru Hasunuma

TL;DR

This work introduces and analyzes the exponential graph operation G^H as a novel graph construction motivated by very large-scale networks. It provides a precise definition, derives fundamental structural properties, and establishes how diameter and connectivity behave under exponentiation, including an exact diameter formula $\mathrm{diam}(G^H) = \mathrm{diam}(G) \cdot |V(H)| + \mathrm{diam}^*(H)$ and a key bound framework for $\mathrm{diam}^*(H)$. The paper proves that every connected exponential graph is maximally connected and gives necessary and sufficient conditions for super edge-connectedness, along with several sufficient conditions ensuring Hamiltonicity, edge-disjoint Hamiltonian cycles, and completely independent spanning trees. By applying these results, the authors construct multi-exponential networks with explicit order and logarithmic diameter (and compare them to the DCell network), and introduce iterated exponential graphs (e.g., $\Omega_k$, $\Psi_k$) that may be of practical and theoretical interest. The work thus advances network design theory by offering a versatile operation to generate high-connectivity graphs with scalable, controlled diameter properties.

Abstract

Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that every connected exponential graph is maximally connected. For exponential graphs, we also present a necessary and sufficient condition to be super edge-connected and sufficient conditions to be Hamiltonian and to have edge-disjoint Hamiltonian cycles and completely independent spanning trees. Applying our results to previously known networks, we have maximally connected and super edge-connected Hamiltonian graphs of doubly exponential order with logarithmic diameter. We furthermore define iterated exponential graphs which may be of not only practical but also theoretical interest.

Exponentiation of Graphs

TL;DR

This work introduces and analyzes the exponential graph operation G^H as a novel graph construction motivated by very large-scale networks. It provides a precise definition, derives fundamental structural properties, and establishes how diameter and connectivity behave under exponentiation, including an exact diameter formula and a key bound framework for . The paper proves that every connected exponential graph is maximally connected and gives necessary and sufficient conditions for super edge-connectedness, along with several sufficient conditions ensuring Hamiltonicity, edge-disjoint Hamiltonian cycles, and completely independent spanning trees. By applying these results, the authors construct multi-exponential networks with explicit order and logarithmic diameter (and compare them to the DCell network), and introduce iterated exponential graphs (e.g., , ) that may be of practical and theoretical interest. The work thus advances network design theory by offering a versatile operation to generate high-connectivity graphs with scalable, controlled diameter properties.

Abstract

Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that every connected exponential graph is maximally connected. For exponential graphs, we also present a necessary and sufficient condition to be super edge-connected and sufficient conditions to be Hamiltonian and to have edge-disjoint Hamiltonian cycles and completely independent spanning trees. Applying our results to previously known networks, we have maximally connected and super edge-connected Hamiltonian graphs of doubly exponential order with logarithmic diameter. We furthermore define iterated exponential graphs which may be of not only practical but also theoretical interest.
Paper Structure (4 sections, 8 theorems, 19 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exponential Graphs
  3. Diameter
  4. Connectivity

Key Result

Proposition 1

Let $G$ and $H$ be graphs of orders $p$ and $q$, respectively. Then,

Figures (1)

  • Figure 1: The Cartesian product graph $C_8^{[2]}$ and the exponential graph $C_8^{K_2}$.

Theorems & Definitions (13)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • ...and 3 more