Consider a regular bipartite graph $G(A,B)$ where the degree of every vertices in $A$ is $d_A$ and in $B$ is $d_B$.
We denote the elements of $A$ by the natural numbers. That is, $A = \lbrace 0, 1, \ldots, |A| - 1 \rbrace$.
An element $b \in B$ is said to be local if it is connected to consecutive elements of $A$. The locality $L(G)$ of a graph is the number of local elements of $B$.
For given $d_A, d_B$ and $|A|$, I want to find a $(d_A, d_B)-$regular graph $G$ that maximise the product $L(G) C(G)$ where $C(G)$ is the length of the smallest cycle in $G$.
It seems hard to find such optimal graph, but I would be happy with any graph that is close to it.
To be honest, I have no idea how to solve this problem except generating random graphs and hoping to find a good solution. Therefore, any help would be appreciate, including good references that can help me.
Thanks!