TL;DR
This paper explores the group-theoretic properties of symmetric graphs and their quotients, examining how additional combinatorial and bipartite structures can help recover original graphs from their quotients.
Contribution
It provides a characterization of arc-transitive graphs via group theory and investigates conditions under which the original graph can be reconstructed from quotient information.
Findings
Identifies combinatorial structures that aid in graph recovery.
Provides criteria for when quotient information suffices for reconstruction.
Enhances understanding of symmetry and quotient relationships in graphs.
Abstract
In this expository paper we describe a group theoretic characterization of arc-transitive graphs and their quotients. When passing from a symmetric graph to its quotient, much information is lost, but some of this information may be recovered from a certain combinatorial design on the blocks, as well as a bipartite graph between the blocks. We address the "extention problem" which asks, when is this additional information sufficient to recover the original graph?
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Coding theory and cryptography