The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, δ ≤ Z(G) where δ is the minimum degree, in the triangle-free case. In particular, we show that 2δ − 2 ≤ Z(G) for graphs with girth of at least 5, and this can be further improved when G has a small cut set. Lastly, we make a conjecture that the lower bound for Z(G) increases as a function of the girth, g, and δ.

We present a counterexample to a lower bound for the power domination number given in Liao, Power domination with bounded time constraints, J. Comb. Optim. 31 (2016): 725-742. We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al, Propagation time for zero forcing on a graph, Discrete Appl. Math., 160 (2012Math., 160 ( ): 1994Math., 160 ( -2005

No abstract