TITLE: Constructing an Infinite Family of Graphs
QUESTION [5 upvotes]: In a number of graph theory books that I am working through I come across questions that ask something like $\textit{'Construct an infinite}$ $\textit{family of graphs with _________'}$  and state some property in the blank space that the family of graphs have to satisfy. These questions totally stump me, so my question is what, if any, is a good method of attack for these types of questions? Also what is a 'solution' to these questions meant to look like?
As an example, I am currently trying to work through the following problem from John Meier's book $\textit{Groups, Graphs and Trees; An Introduction to the Geometry of Infinite Groups}$ that asks:

Construct infinitely many distinct (2,3)-biregular graphs

In this case I understand what a biregular graph is and can construct a number of distinct graphs by hand, however, I don't understand how to generalise it to get infinitely many distinct graphs of this type.

REPLY [5 votes]: The question in your first paragraph is way too broad; it's almost like asking "How do I find a graph that satisfies some given conditions?" The difference is, you're asking "How do I find a graph that satisfies some given conditions, where one of the conditions is 'has more than $n$ vertices'?" The question is unanswerable because what gives the problem its flavor is not the part about having more than $n$ vertices, it's the other conditions. If you ask such a general question, the best you can expect is a very general answer, like "Think hard about it."
Your question about $(2,3)$-biregular graphs is the kind we can deal with here. As stated, it's too easy: just consider the graph $nK_{3,2}$, the union of $n$ vertex-disjoint copies of the complete bipartite graph $K_{3,2}$. To make it a little more interesting, lets ask for large connected $(2,3)$-biregular graphs. There are still lots of solutions. Here's one:
Draw an $n\times n$ chessboard, $n\ge3$. Divide each little square into two triangles by drawing a diagonal. Make the chessboard into a torus by identifying the left side with the right side and the top with the bottom. Now each of the $2n^2$ triangles has $3$ sides, and each of the $3n^2$ edges is a side of $2$ triangles, so the triangle-edge incidence graph is a $(3,2)$-biregular graph of order $5n^2$.