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    Title: Unoriented Laplacian maximizing graphs are degree maximal
    Authors: 譚必信;Tam, Bit-shun;Fan, Yi-zheng;Zhou, Jun
    Contributors: 淡江大學數學學系
    Keywords: unoriented Laplacian matrix;spectral radius;perron vector;maximizing;maximal graph;threshold graph;degree sequence;vicinal pre-order
    Date: 2008-08
    Issue Date: 2011-10-01 21:12:44 (UTC+8)
    Publisher: Philadelphia: Elsevier Inc.
    Abstract: A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices it, v of G, the sets N(u) \ {v}, N(v) \ {u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed. (C) 2008 Elsevier Inc. All rights reserved.
    Relation: Linear Algebra and its Applications 429(4), pp.735-758
    DOI: 10.1016/j.laa.2008.04.002
    Appears in Collections:[數學學系暨研究所] 期刊論文

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