on the reciprocal distance laplacian spectral radius of graphs

The reciprocal distance laplacian matrix of a connected graph $g$ is defined as $rd^l(g)=rtr(g)-rd(g)$, where $rtr(g)$ is the diagonal matrix whose $i$-th element $rtr(v_i)=sum_{ine jin v(g)} frac{1}{d_{ij}}$ and $rd(g)$ is the harary matrix. $rd^l(g)$ is a real symmetric matrix and we denote its eigenvalues as $lambda_1(rd^l(g))geq lambda_2(rd^l(g))geqdotsgeqlambda_n(rd^l(g))$. the largest eigenvalue $lambda_1(rd^l(g))$ of $rd^l(g)$ is called the reciprocal distance laplacian spectral radius. in this paper, we obtain upper bounds for the reciprocal distance laplacian spectral radius. we characterize the extremal graphs attaining this bound.