Improved lower bounds for the degree-diameter problem from metacyclic groups, coset graphs and affine lifts

We present graphs that improve the best published lower bounds for the degree-diameter problem in eighteen parameter pairs $(d,k)$ with degree $d \le 20$ and diameter $k \le 10$. Fifteen of the graphs are Cayley graphs of semidirect products of two cyclic groups, given by explicit generating sets. One is a coset graph of a finite simple group: a $6$-regular graph of diameter $5$ on $1518$ vertices from $\mathrm{PSL}(2,23)$. The remaining two, of degree $20$ and diameters $3$ and $5$ on $2750$ and $450000$ vertices, come from a lift construction in which a small regular graph is blown up by fibres $F_q^{\,s}$ and each edge joins a fibre point to an affine line in the neighbouring fibre; the diameter-$5$ graph exceeds the previous bound by $59.7$ percent. Every graph is specified exactly, and the claimed degree and diameter of each are confirmed by independent breadth-first search, from every vertex when the graph carries no transitive symmetry. Complete descriptions, adjacency data and standalone verification scripts accompany the paper.