Quantum graphs—networks composed of vertices connected by edges on which quantum wave dynamics are defined—have emerged as a versatile model for exploring the interplay between geometry, topology, and ...

When the mathematicians Jeff Kahn and Gil Kalai first posed their “expectation threshold” conjecture in 2006, they didn’t believe it themselves. Their claim — a broad assertion about mathematical ...

For any α ∈ (0, 1) and any nα ≤ d ≤ n/2, we show that λ(G) ≤ Cα√d with probability at least 1− 1 n , where G is the uniform random undirected d-regular graph on n vertices, λ(G) denotes its second ...

Discrete structures are omnipresent in mathematics, computer science, statistical physics, optimisation and models of natural phenomena. For instance, complex random graphs serve as a model for social ...

As mathematical abstractions go, graphs are among the simplest. Scatter a bunch of points in a plane. Connect some of them with lines. That’s all a graph is. And yet they are incredibly powerful. They ...