Few doubts in the graph of group

Question

Let $G$ be any finite group. Let $\Gamma_G$ be a graph corresponding to the group $G$. The vertices of $\Gamma_G$ corresponds to conjugacy classes of group $G$; however not all conjugacy classes of $G$ are vertices of $\Gamma_G$. For a connected component $A$ of $\Gamma_G$, let $\text{Elts} (A)$ denotes the set of all $g \in G$ that are members of some conjugacy class in $A$.

Question 1 : What is edge set of the graph $\Gamma_G$? The number of vertices in the graph will be less than equal to $|G|$.

Question 2 : How to prove that if a decomposition of $G$ contains $t$ non-abelian indecomposable components then the number of connected components in $\Gamma_G$ is atleast $t$?

Answer 1

The definition you are looking for is as follows:

Definition. The graph of a group $G$ (denoted $\Gamma_G$) is a graph with irreducible conjugacy classes as nodes and such that a pair of nodes is connected by an edge if and only if the corresponding pair of conjugacy classes does not commute.

I think this is called a "fusion graph". However, I cannot remember the definition of fusion graphs properly, and I gave up trying to find the definition after a quick google. Perhaps you will have more luck...

The issue here is annoying, but common one made by authors of papers: a lack a sign posting in a paper (the OP links to this paper in the comments, where their question arose). The graph is introduced on page 4 in the "algorithm outline" part of the introduction. It is formally defined on page 10. Ideally, on p4 the authors would have told you to look at p10 for the formal definition of your graph. On the other hand, you should always be wary of using the introduction to understand a paper - sometimes the authors are overly sketchy there.

(The introduction is the most important part of a paper - people read the abstract, and maybe the introduction. I remember walking across a car park with a big shot who screamed "THERE IS NO READER!!!" as, well, no one actually reads your paper, apart from the referees (hopefully). Therefore, signposting in the introduction is important - assume that your reader will not read the whole paper and tell them where to look. Altering their last sentence on p3 to "From the given group $G$, we construct a graph $\Gamma_G$ (defined in Section 6) which has the following properties:..." would make a big difference.)

(Also, because this answer has evolved into some spurious advice on writing papers - avoid using "if and only if" in a definition, unlike in the definition here. It is unnecessary as you are telling me what to do.)