3 problems

Let $\mathcal{G}$ be a graph and let $\beta$-subgraph regular mean that $[e^{\beta A}]_{ii}=[e^{\beta A}]_{jj}$ for all vertices $i,j$. Single-parameter subgraph-regularity conject…

Let $\mathcal{G}$ be a graph. It is walk-regular if $[A^r]_{ii}=[A^r]_{jj}$ for all vertices $i,j$ and every positive integer $r$, and it is $\beta$-subgraph regular if…

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…