A cycle power graph 
is the 
th
graph power of the cycle
graph 
(Louis 2017). Thus, two vertices are adjacent in 
exactly when their graph
distance in 
is at most 
.
Equivalently, 
is the circulant graph 
.
For 
,
the cycle power graph 
is a regular graph of degree 
and has 
edges. For 
, it is the complete
graph 
.
Some standard graph families occur as cycle power graphs as follows.
| graph | cycle power | range |
cycle graph  |  |  |
 -antiprism graph |  |  |
cocktail party graph  |  |  |
complete graph  |  |  |
Vasilev (2026, Theorem 1) proved that for integers 
, 
, and 
, the maximum number of edges in a 
-vertex vertex-induced
subgraph of 
is attained by any set of 
consecutive vertices of the underlying cycle. For fixed 
and sufficiently large 
, Makai et al. (2025, Lemmas 10 and 11) independently
obtained the same consecutive-vertex extremal conclusion under more restrictive hypotheses.
Writing 
for the number of edges in a graph 
, Vasilev's theorem gives
![max_(U subset= V(C_n^r); |U|=k)e(C_n^r[U])={(k; 2) for r>=|_n/2_|; e(P_k^r) for r<|_n/2_| and k<=n-r; e(P_k^r)+(k-n+r+1; 2) for r<|_n/2_| and k>n-r ,](/images/equations/CyclePowerGraph/NumberedEquation1.svg) |
(1)
|
where 
is the 
th
power of the path graph 
and
 |
(2)
|
Vasilev's extremal result also gives the Cheeger constant 
for 
.
The 
-antiprism graph is the special case 
, so its Cheeger constant is 
for 
. For 
, 
is a complete graph,
whose Cheeger constant is ![[n/2]](/images/equations/CyclePowerGraph/Inline48.svg)
.
The consecutive vertex sets need not be the only maximizers. For example, label the vertices of the underlying cycle graph 
as 1, 2, ..., 6 in cyclic order. The vertices numbered 1,
3, and 5 then form a triangle in 
, so this nonconsecutive vertex set also induces the maximum
of three edges. A classification of all maximizing vertex sets remains open (Vasilev
2026).
