New graphs related to (p,6) and (p,8)-cages

Constructing regular graphs with a given girth, a given degree and the fewest possible vertices is hard. This problem is called the cage graph problem and has some links with the error code theory. $G$ ‐graphs can be used in many applications: symmetric and semi‐symmetric graph constructions, (Bretto and Gillibert (2008) ), hamiltonicity of Cayley graphs, and so on. In this paper, we show that $G$ ‐graphs can be a good tool to construct some upper bounds for the cage problem. For $p$ odd prime we construct $(p,6)$ ‐graphs which are $G$ ‐graphs with orders $2p^2$ and $2p^2-2$ , when the Sauer bound is equal to $4{(p-1)}^3$ . We construct also $(p,8)$ ‐ $G$ ‐graphs with orders $2p^3$ and $2p^3-2p$ , while the Sauer upper bound is equal to $4{(p-1)}^5$ .