On the Faria's inequality for the Laplacian and signless Laplacian spectra: A unified approach

Let p(G) and q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m(L)+/-((G)) (1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that m(L) +/-((G))(1) is bounded below by p(G) - q(G). Let r(G) be the number of internal vertices of G. If r(G) = q(G), following a unified approach we prove that m(L) +/- ((G)) (1) = p(G) - q(G). If r(G) > q(G) then we determine the equality m(L) +/- ((G)) (1) = p(G) - q(G)-m(N) +/- (1), where m(N) +/- (1) denotes the multiplicity of 1 as eigenvalue of a matrix N-+/-. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are nonquasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs. (C) 2015 Elsevier Inc. All rights reserved.