I would like to ask. I found the original image from Wolfram website and then cropped this example of circulant graph. Source: https://mathworld.wolfram.com/QuarticGraph.html
Why does it says it is circulant graph (1,2)? Should not it be (1,5)? I am new to graph theory, so would like to make sure, and I dont see in google about that notation also. Thanks in advance.
On
One of the definitions of a circulant graph: The graph is a Cayley graph of a cyclic group.
For example, $\operatorname{Cay}(\mathbb{Z}_n;a_1,\ldots,a_r)$ is a Cayley graph of $\mathbb{Z}_n$ with the set of generators consisting of elements $\{a_1,\ldots,a_r\}$.
Thus the $11$-circulant graph $(1,2)$ is the $\operatorname{Cay}(\mathbb{Z}_{11};1,2)$, similarly the $11$-circulant graph $(1,5)$ is $\operatorname{Cay}(\mathbb{Z}_{11};1,5)$.
Since $\operatorname{Cay}(\mathbb{Z}_{11};1,2)\cong \operatorname{Cay}(\mathbb{Z}_{11};1,5)$ (the second graph is obtained from the first one by multiplication by $5$), of course we can assume that the picture shows both mentioned graphs.
The picture is "correct", but it is drawn very confusingly, and you are right to be suspicious. The clearer version of the picture would have an 11-cycle around the outside, with an edge between every pair distance 2 on the cycle (which is what I think you were expecting to see).
The vertices of distance $1$ apart are connected by the long, diagonal edges. In particular, these form a cycle of length 11. The vertices distance $2$ apart (as in, distance two apart on the cycle formed by the long edges) are connected by short edges. So it is $C(1,2)$.
EDIT: The notation works like this. The $n$-Circulant graph $C(a_1, a_2, \dots, a_k)$ has vertex set $\{0,1,2,3,\dots, n-1\}$. You make two vertices $i$ and $j$ adjacent whenever the difference $|i-j| = a_m \pmod n$, for some $a_m$ in $\{a_1, a_2, \dots, a_k\}$.
So in this example, your vertex set is $\{0, 1, \dots, 10\}$, and any time two vertices differ by either $1$, or $2$, modulo $11$, you make them adjacent.