Regularity of symbolic powers of certain graphs

Let $G_{n,r}$ denote the graph with $n$ vertices $\{x_1,\ldots,x_n\}$ in cyclic order and for each vertex $x_i$ consider the set $A_i=\{x_{i-r},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots, x_{i+r}\},$ where $x_{i-j}$ is the vertex $x_{n+i-j}$, whenever $i<j$ and $0\leq r\leq \Bigl\lfloor\dfrac{n}{2}\Bigr\rfloor -1$. In $G_{n,r}$, every vertex $x_i$ is adjacent to all the vertices of $V(G_{n,r})\backslash A_i$. Let $I=I(G_{n,r})$ be the edge ideal of $G_{n,r}$. We show that Minh's conjecture is true for $I,$ i.e. regularity of ordinary powers and symbolic powers of $I$ are equal. We compute the Waldschmidt constant and resurgence for the whole class.

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