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

Given a connected graph G , the distance Laplacian matrix D L (G) is defined as D L (G) = Tr (G) − D (G) , and the distance signless Laplacian matrix D Q (G) is defined as D Q (G) = Tr (G) + D (G) , where Tr (G) is the transmission matrix of G and D (G) is the distance matrix of G. The Indu–Bala product of two graphs G 1 and G 2 , denoted by G 1 ▾ G 2 , 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 G 1 ▾ G 2 in terms of Laplacian spectra of G 1 and G 2 . We then obtain the distance signless Laplacian spectrum of G 1 ▾ G 2 in terms of signless Laplacian spectra of G 1 and G 2 . 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 G 1 ▾ G 2 in terms of metric dimensions of G 1 and G 2 . Finally, we provide a problem for future research. [ABSTRACT FROM AUTHOR]