A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge lengths. We show that the radical solvability of a generic framework depends only on its underlying graph and characterise which planar graphs give rise to ...

Since the introduction of LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al., 2000), a large number of non-linear dimensionality reduction techniques (manifold learners) have been proposed. Many of these non-linear techniques can be viewed as instantiations of Kernel PCA; they employ a cleverly designed kernel matrix that preserves local data structure in the “feature space” (Bengio et al...

Abstract: Let G be a graph. The partially squared graph G∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv 6∈ E(G), and there is some w ∈ N(u) ∩ N(v), such that N(w) ⊆ N(u) ∪ N(v) ∪ {u, v}. Let t > 1 be an integer and Y ⊆ V (G), denote n(Y ) = |{v ∈ V (G) | min y∈Y {distG(v, y)} ≤ 2}|, It(G) = {Z |Z is an independent set of G, |Z| = t}. In this paper, we show tha...

The d-measurement set of a graph is its set of possible squared edge lengths over all d-dimensional embeddings. In this note, we define a new notion of graph isomorphism called d-measurement isomorphism. Two graphs are d-measurement isomorphic if there is agreement in their d-measurement sets. A natural question to ask is “what can be said about two graphs that are d-measurement isomorphic?” In...

It is shown that there exists a sequence of 3-regular graphs {Gn}n=1 and a Hadamard space X such that {Gn}n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn}n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approxima...