On eigenvalues of the reciprocal distance signless Laplacian matrix of graphs

Abstract

For a simple connected graph [Formula: see text], the reciprocal transmission [Formula: see text] of a vertex [Formula: see text] is defined as [Formula: see text] The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Harary matrix (reciprocal distance matrix) of [Formula: see text] and [Formula: see text] is the diagonal matrix of the vertex reciprocal transmissions in [Formula: see text]. In this paper, we first investigate the RDSL spectrum of the join and also joined union of graphs and we construct new pairs of equienergetic graphs by using the joined union of regular graphs. Then we obtain some bounds on the spectral radius of reciprocal distance signless Laplacian matrix of [Formula: see text], which is the largest eigenvalue of the matrix [Formula: see text]. We also characterize the extremal graphs.