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グラフスペクトルによる多項式時間最大クリーク抽出についてのプレゼン資料

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The maximum clique problem (MCP) is to determine a sub graph of maximum cardinality. A clique is a sub
graph in which all pairs of vertices are mutually adjacent. Based on existing surveys, the main goal of this paper is to
provide a simplified version and comprehensive review on Maximum clique problem. This review intends to encourage
and motivate new researchers in this area. Though capturing the complete literature in this regard is beyond scope of the
paper, but it is tried to capture most of the representative papers from similar approaches.

We give several old and some new applications of eigenvalue interlacing to matrices associated to graphs. Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (co)clique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity results concerning the structure of graphs and block designs.

Part I contained a description of the single-modulus algorithm for reducing a matrix to Frobenius form, obtaining exact integral factors of the characteristic polynomial. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial. Part II also contains a discussion of the selection of the moduli and numerical examples.

A description is given of the Danilewski algorithm for reducing a matrix
to Frobenius form using rational arithmetic. This algorithm is modified for use
over the field of integers modulo p. The modified algorithm yields exact integral
factors of the characteristic polynomial. A description of the single-modulus
algorithm is given. Part II contains a description of the multiplemodulus
algorithm. Since different moduli may yield different factorizations, an
algorithm is given for determining which factorizations are not correct
factorizations over the integers of the characteristic polynomial. (auth)

- X.-D Zhang

X.-D. Zhang.
The laplacian eigenvalues of graphs: a survey.
arXiv preprint
arXiv:1111.2897, 2011.