Minimal Isometric Embeddings of Graphs into Abelian Groups: Theory, Algorithms, and Applications to Signal Processing over Networks

Abstract

This dissertation develops a framework for embedding arbitrary connected graphs isometrically into Cayley graphs of abelian groups, with applications to harmonic analysis on networks. It addresses representing irregular graph-structured data within highly symmetric algebraic hosts, on which classical Fourier theory applies verbatim rather than by analogy. The theoretical core is twofold. First, we introduce edge relations phi, Phi, and Psi that detect metric parallelism, a strict generalization of the Djokovic-Winkler relation beyond bipartite and partial-cube structures, with a transitive prune operation converting them into candidate same-generator edge partitions. Second, we prove the Cocycle/Quotient Labeling Theorem: any edge partition induces a most-generic consistent vertex labeling as a GF(2) quotient of dimension k = t - rank(A), where A is the cycle-class parity matrix; the labeling can fail only by shortcuts, never by stretching. With a shortcut-repair loop terminating in the isometric spanning-tree embedding, this gives a universal algorithm: every connected graph G embeds isometrically into a Cayley graph of (Z_2)^k with k <= n-1, verified exhaustively on all 995 connected graphs of at most seven vertices. A bounds theory follows: k >= max(diam(G), ceil(log2 n)); stars satisfy k_min(K_{1,q}) = ceil(log2 q) + 1, exponentially below the naive dimension; odd cycles require k = n-1. We then generalize the quotient machinery from GF(2) to Z via the Smith Normal Form, giving embeddings into products of cyclic groups. The primary application is harmonic analysis: these embeddings ground Fourier analysis, convolution, and wavelet transforms on graph signals, preserving translation-modulation duality, convolution theorems, and Plancherel identities that matrix-based graph signal processing lacks. We name this framework Group-Embedding-based Graph Signal Processing (GE-GSP).