ON DISTANCE SIGNLESS LAPLACIAN ESTRADA INDEX AND ENERGY OF GRAPHS

For a connected graph G, the distance signless Laplacian matrix is defined as D-Q (G) = Tr(G)+D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of vertex transmissions of G. The eigenvalues rho(1), rho(2), ..., rho(n) of D-Q(G) are the distance signless Laplacian eigenvalues of the graph G. In this paper, we define the distance signless Laplacian Estrada index of the graph G as (DEE)-E-Q(G) = Sigma(n)(i=1) e( (rho i - 2 sigma(G)/n)), where sigma (G) is the transmission of a graph G. We obtain upper and lower bounds for (DEE)-E-Q(G) and the distance signless Laplacian energy in terms of other graph invariants. Moreover, we derive some relations between (DEE)-E-Q(G) and the distance signless Laplacian energy of G.