  A strict lower bound for the diameter of a symmetric graph is proposed, which
is calculable with the order ($n$) and other local parameters of the graph such
as the degree, even girth $g$ ($\geq 4$), and number of $g$-cycles traversing a
vertex, which are easily determined by inspecting a small portion of the graph
(unless the girth is large). It is applied to the symmetric Cayley graphs of
the Rubik's Cube groups of various sizes and metrics, yielding slightly tighter
lower bounds than those of random $k$-regular graphs proposed by Bollob\'{a}s
and de la Vega, which range from 60% to 77% of the correct diameters of
large-$n$ graphs.
