Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

On the spectral radius and energy of signless Laplacian matrix of digraphs

Let D be a digraph of order n and with a arcs. The signless Laplacian matrix Q(D) of D is defined as Q(D)=Deg(D)+A(D), where A(D) is the adjacency matrix and Deg(D) is the diagonal matrix of vertex out-degrees of D. Among the eigenvalues of Q(D) the eigenvalue with largest modulus is the signless Laplacian spectral radius or the Q-spectral radius of D. The main contribution of this paper is a series of new lower bounds for the Q-spectral radius in terms of the number of vertices n, the number of arcs, the vertex out-degrees, the number of closed walks of length 2 of the digraph D. We characterize the extremal digraphs attaining these bounds. Further, as applications we obtain some bounds for the signless Laplacian energy of a digraph D and characterize the extremal digraphs for these bounds.

On the Generalized Distance Energy of Graphs

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.