Let $G=(V,E)$ be a connected graph with $|E|=17$ and for all vertices $\deg(v)>3$. What is the maximum value of $|V|$? Let $G=(V,E)$ be a connected graph with $|E|=17$ and for all vertices $\deg(v)>3$. What is the maximum value of $|V|$? (What is the maximum possible number of vertices?)

There are $24$ non-isomorphic $8$-vertex $17$-edge connected graphs with minimum degree $\geq 4$.  They can be generated using geng which comes with nauty using the command:
geng 8 17:17 -d4 -c

I wrote a script to display the output.  Here's the result:

The first one is $K_{4,4}$ with an additional edge.
By the way, if we attempt
geng 9 17:17 -d4 -c

we receive the response:
>E geng: impossible mine,maxe,mindeg,maxdeg values

which gives a computer verification that $8$ is indeed the maximum value.
