On the reduced signless Laplacian spectrum of a degree maximal graph

✍ Scribed by Bit-Shun Tam; Shu-Hui Wu

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 314 KB
• Volume: 432
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.11.031

✦ Synopsis

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 (G) + B(G), where B(G) is the reduced adjacency matrix of G and (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 wellknown 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.