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

Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G  , respectively. Let mL±(G)(1)mL±(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 mL±(G)(1)mL±(G)(1) is bounded below by p(G)−q(G)p(G)−q(G). Let r(G)r(G) be the number of internal vertices of G  . If r(G)=q(G)r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G)mL±(G)(1)=p(G)−q(G). If r(G)>q(G)r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1)mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1)mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.