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.

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Spectral Graph Theory and its Applications

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.

Algebraic Complexity Theory.

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.

Canonical labeling of graphs

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.

Quantum algorithms for algebraic problems

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.

Engineering an efficient canonical labeling tool for large and sparse graphs

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.

https://hal.archives-ouvertes.fr/hal-03053174/document

D. Yu. Grigoryev

Papers

Isomorphism of graphs with bounded eigenvalue multiplicity