Exponentiation of Graphs

TL;DR

This paper introduces the concept of graph exponentiation, enabling the construction of large, highly connected networks with desirable properties like logarithmic diameter and Hamiltonicity, relevant for large-scale communication systems.

Contribution

It defines graph exponentiation, characterizes properties of exponential graphs, and applies these to construct highly connected, large-scale network models with specific connectivity and Hamiltonian properties.

Findings

Exponential graphs can have logarithmic diameter.

Connected exponential graphs are maximally connected.

Conditions for super edge-connected and Hamiltonian properties are established.

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.