–Rényi (ER) random graphs, with z neighbors for each spin. In the case with spin S=1/2, the undirected and directed ER graphs … graphs the model presents a spontaneous magnetization at p=z/N(z=2,3,…,N), but no spontaneous magnetization at p=1/N which is … the percolation threshold. For both directed and undirected ER graphs with spin S=1, we find a first-order phase …

We study the connected regular graphs with four distinct eigenvalues. Properties and feasibility conditions of the …

equally large sets at maximal distance are given, and we find graphs for which the bounds are tight. …

this is the case.As a consequence we find (among others) that the following distance-regular graphs are uniquely determined … by their spectrum: The collinearity graphs of the generalized octagons of order (2; 1), (3; 1) and (4; 1), the Biggs …-Smith graph, the M22 graph, and the coset graphs of the doubly truncated binary Golay code and the extended ternary Golay code. …