The spectrum and metric dimension of Indu–Bala product of graphs

Given a connected graph [Formula: see text], the distance Laplacian matrix [Formula: see text] is defined as [Formula: see text], and the distance signless Laplacian matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the transmission matrix of [Formula: see text] and [Formula: see text] is the distance matrix of [Formula: see text]. The Indu–Bala product of two graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], was introduced in (G. Indulal and R. Balakrishnan, Distance spectrum of Indu–Bala product of graphs,  AKCE Int. J. Graph Comb. 13(3) (2016) 230–234). In this paper, we first obtain the distance Laplacian spectrum of [Formula: see text] in terms of Laplacian spectra of [Formula: see text] and [Formula: see text]. We then obtain the distance signless Laplacian spectrum of [Formula: see text] in terms of signless Laplacian spectra of [Formula: see text] and [Formula: see text]. We construct pair of graphs which are distance Laplacian co-spectral as well as pair of graphs which are distance signless Laplacian co-spectral. We further find the metric dimension of [Formula: see text] in terms of metric dimensions of [Formula: see text] and [Formula: see text]. Finally, we provide a problem for future research.