On the reduced signless Laplacian spectrum of a degree maximal graph

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.