for integer m, $S \subseteq \{1,2,3...m-1\}$ and undirected graph $G_{m,s} =(V_m, E_{m,s})$ with definition below
$V_{m} ={1,2,...m}$
$E_{m,S}=\{{u,v}|((u-v) mod m) \in S\}$
I need to find equation of diameter of $G_{m,\{1,2,..k\}}$ where $k \leq \frac{m}{2}$ using m and k
From the problem I know that there is edge in graph if $u,v$ have same remainder modulo n, which mean is u-v is multiple of n ,
for example $G_{6,\{2\}}$ means vertices is {1,2...6} and edges $(u-v) $ mod 2 $\in S$ so we can partition $ 1 \equiv 3 \equiv 5 $ mod 2 and $ 2 \equiv 4 \equiv 6$ mod 2
Edges = $\{\{1,3\},\{3,1\}\{4,2\},\{2.4\},\{3,5\},\{5,3\},\{6,4\},\{4,6\},\{6,2\}\{2,6\},\{1,5\},\{5,1\}\}$
but how can i conclude and proof diameter of graph?
what i think is that $G_{m,\{1,2,..k\}}$ and $k \leq \frac{m}{2}$ have edge in every connected vertex so radius and diameter is 1, because if i draw the graph there is edge connected all vertices