The graph isomorphism problem: its structural complexity
The graph isomorphism problem: its structural complexity
On determining the congruence of point sets in d dimensions
Computational Geometry: Theory and Applications
Testing the congruence of d-dimensional point sets
Proceedings of the sixteenth annual symposium on Computational geometry
The Complexity of Modular Graph Automorphism
SIAM Journal on Computing
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
Computing Largest Common Point Sets under Approximate Congruence
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Isomorphism of graphs with bounded eigenvalue multiplicity
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms and Data Structures: The Basic Toolbox
Algorithms and Data Structures: The Basic Toolbox
Simplices and Spectra of Graphs
Discrete & Computational Geometry
Proofs from THE BOOK
The Laplacian lattice of a graph under a simplicial distance function
European Journal of Combinatorics
| Hi-index | 5.23 |
Given a connected undirected graph, we associate a simplex with it such that two graphs are isomorphic if and only if their corresponding simplices are congruent under an isometric map. In the first part of the paper, we study the effectiveness of a dimensionality reduction approach to Graph Automorphism. More precisely, we show that orthogonal projections of the simplex onto a lower dimensional space preserves an automorphism if and only if the space is an invariant subspace of the automorphism. This insight motivates the study of invariant subspaces of an automorphism. We show the existence of some interesting (possibly lower dimensional) invariant subspaces of an automorphism. As an application of the correspondence between a graph and its simplex, we show that there are roughly a quadratic number of invariants that uniquely characterize a connected undirected graph up to isomorphism. In the second part, we present an exponential sum formula for counting the number of automorphisms of a graph and study the computation of this formula. As an application, we show that for a fixed prime p and any graph G, we can count, modulo p, the number of permutations that violate a multiple of p edges in G in polynomial time.