I am not sure if this is what you are looking for, but I'll try......(It would have been better, if you'd mentioned what you understood, and what you're looking for.)
First of all, you should know the difference between a continuous variable and a discrete variable.
A continuous variable is a variable, which can take an infinity of values between any two chosen points. A discrete variable, on the other hand is one, which can take only a fixed number of values, between any two chosen points.
The process of discretization involves converting a continuous variable in to a discrete variable. This has the advantage of reducing the computational complexity of the resulting model(fewer values mean fewer calculations). The exact process of how it is done, depends on the phenomenon we're modelling.In engineering applications, for example it involves discarding values that are insignificant (i.e which introduce only a small error, when we physically realize the model).
There are two key things to keep in mind while creating a discrete model -
- Accuracy 2. Computational efficiency
As you can imagine, we have to maintain an optimal tradeoff between the two. Increasing one will decrease the other. The bulk of the process of discretization, centers on this optimization process.
The meaning of the word accuracy needs to be explained here - you are using the discrete model as a substitute for the continuous(original) model. So we should be able to extract the main features of the original model from the discrete model. (To be able to do this while retaining computational efficiency is where the ability of the mathematician/engineer/scientist who's creating the model, lies.)
This is a general explanation of discretization, without any specific field of application in mind.(the engineering reference was because of my own background.)
I hope it helps.... if not, please let me know.
I can't explain this approach to a five-year-old child. But here are my considerations, which work for any graph, but perhaps not very effectively from the computer science point of view.
Any graph of order $n$ has an empty subgraph and $n$ single-vertex subgraphs.
More generally, there are $n\choose k$ possibilities to choose $k$ vertices from $n$.
If a graph of order $k$ has $m$ edges, then it has $2^m$ different subgraphs of order $k$.
Applying considerations 3 and 4 for each $k=2,\ldots,n$ we can compute the total number of subgraphs of a given graph.
Addition. For example, for the graph in this post
How many subgraphs of this simple graph? it looks like this:
empty subgraphs $(1)$
subgraphs with a single vertex $(4)$
subgraphs with two vertices - $6$ of them
2a. three have no edges $(3)$
2b. three have exactly one edge $(3\cdot2)$
subgraphs with three vertices - $4$ of them
3a. one has no edges $(1)$
3b. three have exactly two edges $(3\cdot2^2)$
subgraphs with four vertices $1$. It has three edges ($2^3$)
Now let's sum up all the numbers in brackets
$$
1+4+3+3\cdot2+1+3\cdot2^2+2^3=35.
$$
Best Answer
The paper you linked to answers your question. No, generalized fan graphs are not necessarily simple. The fan-type graph $F_{k_1, \dotsc,k_n}$ denotes the graph that is a path with $n$ vertices $\{v_1,\dotsc,v_n\}$ in that order, with the addition of a single new vertex $v_0$ having $k_i$ edges connecting $v_0$ to $v_i$. Similarly as the paper describes,
It appears that the authors don't define a generalized fan graph, but are just using the term "generalized fan graph" to describe these three types of graphs that are the subject of their paper.