I have a basic question about graph theory. [Side note I did read the other question on here Why is a graph an ordered pair?, it didn't really answer my question.]
Defining a graph as $\\G = (V,E)$. Meaning the graph $\\G$ is made up of the ordered pair $\\(V,E)$. Where $\\V$ is a set of vertices V = {v1,v2,v3 ect} and E is a set of edges E = {{v1,v2} {v2,v3}}.
Lets just also say that I'm talking about an undirected graph to start. As shown by the fact that the elements of E are not ordered pairs.
I'm getting tripped up on how (V,E) seems to be an ordered pair but it doesn't seem like the order of how V and E are listed out matters? Can anyone help me explain why that is? What is the impact of (V,E)?
Not just graphs, but many other mathematical objects are defined as ordered tuples. Examples include groups, rings, topological spaces. Each object comes with parts that serve a particular function. The actual order doesn't matter as long as it is consistent. Having said this, some of the parts depend on other parts. In the graph example, the edges are defined in terms of the vertices, so it seems natural to place the vertices first.
The same problem arises in computer languages with multi-argument functions. What order should they be in? Some computer languages solve that problem by allowing $\texttt{keyword=value}$ pairs. For example, using this idea, a graph is defined by $\;G = (\texttt{vertices}=V,\texttt{edges}=E).\;$ Ultimately, it is just a matter of established convention how it is should be written.