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We introduce the set G SSP of all simple graphs G with the property that each symmetric matrix corresponding to a graph G ∈ G SSP has the strong spectral property. We find several families of graphs in G SSP and, in particular, characterise the trees in G SSP. [ABSTRACT FROM AUTHOR]

We conjecture that all connected graphs of order n have von Neumann entropy at least as great as the star K 1 , n − 1 and prove this for almost all graphs of order n . We show that connected graphs of order n have Rényi 2-entropy at least as great as K 1 , n − 1 and for α > 1 , K n maximizes Rényi α -entropy over graphs of order n . We show that adding an edge to a graph can lower its von Neumann entropy. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]