On the reduced signless Laplacian spectrum of a degree maximal graph |
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Authors: | Bit-Shun Tam Shu-Hui Wu |
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Affiliation: | a Department of Mathematics, Tamkang University, Tamsui 251, Taiwan, ROC b Center for General Education, Taipei College of Maritime Technology, Taipei 111, Taiwan, ROC |
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Abstract: | For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges. |
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Keywords: | 05C50 15A18 |
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