3 problems

Kloster–Král–Sullivan's conjecture. The graph $\mathcal{G}$ is walk regular if and only if there exists a rational $\beta>0$ such that all vertices are $\beta$-subgraph equivalent.

Let $\mathcal{G}$ be a graph whose vertices all have the same $1$-subgraph centrality; equivalently, $[e^A]_{ii}$ is independent of the vertex $i$. The degree, closeness, eigenvect…

Let $\mathcal{G}=(V,E)$ be a simple undirected graph with adjacency matrix $A$. For $\beta>0$, the $\beta$-subgraph centrality of vertex $i$ is $[e^{\beta A}]_{ii}$, and…