Common terminology used for lower/upper bounds
The term “tight” has multiple uses, as is mentioned in a comment. However, you seem to be interested in a combinatorial context or on a combinatorial-type problem so I’m going to assume that contexts. Let’s say we are talking about a function, $f(n)$, that bounds $g(n)$. In your case, $g$ is the true number of vertices of the $n^{th}$ graph in your family and $f$ is your bound.


  $f$ is a tight bound for $g$ if there exists an $n$ such that $g(n)=f(n)$.


Notice that $f$ can be tight but still improvable. If you have that $g(n)=n-\epsilon\sin((n-1)/100)$ then $f(n)=n^2$ is a tight bound. It is an upper bound for $g$ and they agree for $n=1$. However, that doesn’t mean it cannot be improved. A better bound might be $f_k(n)=n^{1+k^{-1}}$. This family has the property that $g(n)&lt;f_k(n)&lt;f(n)$ for $n&gt;1$ and $g(1)=f_k(1)+f(1)$.

I am generally not a fan of the term “optimal bound” because it’s usually inaccurate. The optimal bound for a sequence of integers is simply that sequence of integers, so if you have an “optimal bound” you almost always have an expression for the underlying process.

There are contexts in which the previous paragraphs is wrong, however, and you seem to think that you’ve proven that you bound is the best possible. If that’s the case, there aren’t any caveats to that sentence, and what you’ve proven is indeed a bound rather than an expression for the number of vertices in the graph, then “optimal bound” or “best possible bound” is the terminology that I would recommend using.