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Proceedings ArticleDOI

Isomorphism of graphs with bounded eigenvalue multiplicity

TL;DR: Two polynomial time algorithms are described which test isomorphism of undirected graphs whose eigenvalues have bounded multiplicity, if X and Y are graphs of eigenvalue multiplicity m.
Abstract: We investigate the connection between the spectrum of a graph, i.e. the eigenvalues of the adjacency matrix, and the complexity of testing isomorphism. In particular we describe two polynomial time algorithms which test isomorphism of undirected graphs whose eigenvalues have bounded multiplicity. If X and Y are graphs of eigenvalue multiplicity m, then the isomorphism of X and Y can be tested by an O(n4m+c) deterministic and by an O(n2m+c) Las Vegas algorithm, where n is the number of vertices of X and Y.
Citations
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Proceedings ArticleDOI
21 Oct 2007
TL;DR: This tutorial will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.
Abstract: Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.

845 citations


Cites background from "Isomorphism of graphs with bounded ..."

  • ...In this case, Babai, Grigoriev and Mount [4] showed how to test isomorphism in polynomial time....

    [...]

01 Jan 1990
TL;DR: Algebraic complexity theory as mentioned in this paper is a project of lower bounds and optimality, which unifies two quite different traditions: mathematical logic and the theory of recursive functions, and numerical algebra.
Abstract: Publisher Summary This chapter discusses algebraic complexity theory. Complexity theory, as a project of lower bounds and optimality, unites two quite different traditions. The first comes from mathematical logic and the theory of recursive functions. In this, the basic computational model is the Turing machine. The second tradition has developed from questions of numerical algebra. The problems in this typically have a fixed finite size. Consequently, the computational model is based on something like an ordinary computer that however is supplied with the ability to perform any arithmetic operation with infinite precision and that in turn is required to deliver exact results. The formal model is that of straight-line program or arithmetic circuit or computation sequence, more generally that of computation tree.

569 citations

Proceedings ArticleDOI
01 Dec 1983
TL;DR: An algebraic approach to the problem of assigning canonical forms to graphs by computing canonical forms and the associated canonical labelings in polynomial time is announced.
Abstract: We announce an algebraic approach to the problem of assigning canonical forms to graphs. We compute canonical forms and the associated canonical labelings (or renumberings) in polynomial time for graphs of bounded valence, in moderately exponential, exp(n½ + o(1)),time for general graphs, in subexponential, nlog n, time for tournaments and for 2-(n,k,l) block designs with k,l bounded and nlog log n time for l-planes (symmetric designs) with l bounded. We prove some related problems NP-hard and indicate some open problems.

472 citations

Journal ArticleDOI
TL;DR: This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.
Abstract: Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

288 citations


Cites background from "Isomorphism of graphs with bounded ..."

  • ...y special cases of graph isomorphism, such as when the maximum degree is bounded (Luks, 1982), the genus is bounded (Filotti and Mayer, 1980; Miller, 1980), or the eigenvalue multiplicity is bounded (Babai et al., 1982). Furthermore, there are classical algorithms that run in time 2O( √ nlogn) for general graphs (Babai et al., 1983); and in time 2O( 3 √ n) for stronglyregulargraphs(Spielman,1996), whichare suspected...

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Proceedings Article
06 Jan 2007
TL;DR: Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed.
Abstract: The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing state-of-the-art tools.

269 citations

References
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MonographDOI
16 May 1974
TL;DR: In this article, the authors introduce algebraic graph theory and show that the spectrum of a graph can be modelled as a graph graph, and the spectrum can be represented as a set of connected spanning trees.
Abstract: 1. Introduction to algebraic graph theory Part I. Linear Algebra in Graphic Thoery: 2. The spectrum of a graph 3. Regular graphs and line graphs 4. Cycles and cuts 5. Spanning trees and associated structures 6. The tree-number 7. Determinant expansions 8. Vertex-partitions and the spectrum Part II. Colouring Problems: 9. The chromatic polynomial 10. Subgraph expansions 11. The multiplicative expansion 12. The induced subgraph expansion 13. The Tutte polynomial 14. Chromatic polynomials and spanning trees Part III. Symmetry and Regularity: 15. Automorphisms of graphs 16. Vertex-transitive graphs 17. Symmetric graphs 18. Symmetric graphs of degree three 19. The covering graph construction 20. Distance-transitive graphs 21. Feasibility of intersection arrays 22. Imprimitivity 23. Minimal regular graphs with given girth References Index.

2,924 citations

Book
01 Jan 1995
TL;DR: The Spectrum and the Group of Automorphisms as discussed by the authors have been used extensively in Graph Spectra Techniques in Graph Theory and Combinatory Applications in Chemistry an Physics. But they have not yet been applied to Graph Spectral Biblgraphy.
Abstract: Introduction. Basic Concepts of the Spectrum of a Graph. Operations on Graphs and the Resulting Spectra. Relations Between Spectral and Structural Properties of Graphs. The Divisor of a Graph. The Spectrum and the Group of Automorphisms. Characterization of Graphs by Means of Spectra. Spectra Techniques in Graph Theory and Combinatories. Applications in Chemistry an Physics. Some Additional Results. Appendix. Tables of Graph Spectra Biblgraphy. Index of Symbols. Index of Names. Subject Index.

2,119 citations

Book
01 Jan 1980

1,729 citations

Proceedings ArticleDOI
30 Apr 1974
TL;DR: The time bound for planar graph isomorphism is improved to O(|V|) time and the algorithm can be easily extended to partition a set of planar graphs into equivalence classes of isomorphic graphs in time linear in the total number of vertices in all graphs in the set.
Abstract: The isomorphism problem for graphs G1 and G2 is to determine if there exists a one-to-one mapping of the vertices of G1 onto the vertices of G2 such that two vertices of G1 are adjacent if and only if their images in G2 are adjacent. In addition to determining the existence of such an isomorphism, it is useful to be able to produce an isomorphism-inducing mapping in the case where one exists.The isomorphism problem for triconnected planar graphs is particularly simple since a triconnected planar graph has a unique embedding on a sphere [6]. Weinberg [5] exploited this fact in developing an algorithm for testing isomorphism of triconnected planar graphs in O(|V|2) time where V is the set consisting of the vertices of both graphs. The result has been extended to arbitrary planar graphs and improved to O(|V|log|V|) steps by Hopcroft and Tarjan [2,3]. In this paper, the time bound for planar graph isomorphism is improved to O(|V|). In addition to determining the isomorphism of two planar graphs, the algorithm can be easily extended to partition a set of planar graphs into equivalence classes of isomorphic graphs in time linear in the total number of vertices in all graphs in the set. A random access model of computation (see Cook [1]) is assumed. Although the proposed algorithm has a linear asymptotic growth rate, at the present stage of development it appears to be inefficient on account of a rather large constant. This paper is intended only to establish the existence of a linear algorithm which subsequent work might make truly efficient.

379 citations