I agree with caffeinemachine that you may have misunderstood some definitions. In this case, we may be talking about different things and not realize it. I recommend going back to definitions and making sure that your understanding is correct.
With regard to the discussion above...
Joystar1977 said:
Evgeny.Makarov then is it correct to say that Kn doesn't have anything to connect to since it's a single vertex.
Let's recall what I said in post #8:
Evgeny.Makarov said:
There are graphs $K_3$, $K_4$, $K_5$, but there is no such object as $K_n$ for unknown $n$. Strictly speaking, $K_n$ is a function: you give it $n$, it returns you a complete graph on $n$ vertices. E.g., you give it 5, it returns $K_5$
First, note that $K_n$ is not an indivisible name like "pentagon". In consists of two parts: the name of the collection $K$, which probably stands for "complete", and an index $n$, which takes values 1, 2, …. As I said, $K_n$ is a function $K$ that takes an argument $n$; it is more like $K(n)$. It may have been confusing to write Kn in plain text because it looks like an indivisible name. Recall functions, e.g., $f(x)=x^2$: such $f(x)$ is not a single number; for each $x$ is returns its own number.
Yet you continue treating $K_n$ as if it is a concrete graph and keep saying things like it has a single vertex or it has 45 edges.
$K_n$ is not a single graph, it is a whole collection of graphs: $K_1$, $K_2$, $K_3$ and so on. What is true for $K_1$ (it consists of a single vertex) is not true for $K_4$. What is true for $K_5$ (it has 10 edges) is not true for $K_3$. You cannot say "$K_n$ has 10 edges" and stop there because this is neither true nor false. It may be true for some graphs in the $K_n$ family and false for others. In contrast, the phrase "$K_5$ has 10 edges" has a definite truth value (it is true).
Another way to refer to $K_n$ meaningfully is to use phrases "for all $n$" or "for some $n$". Then you are talking about the collection $K_n$ as a whole. For example, it makes sense to say, "
For all $n$, the graph $K_n$ has $n(n-1)/2$ edges". In this one sentence you make an infinite number of claims: $K_1$ has 0 edges, $K_2$ has 1 edge, $K_3$ has 3 edges, $K_4$ has 6 edges and so on. It also makes sense to say, "$K_n$ has 10 edges
for some $n$", namely for $n=5$.
Usually when we refer to $K_n$ without specifying concrete $n$ and without using either of the phrases "for all $n$" or "for some $n$" (or their equivalents), then we implicitly mean "for all $n$". Thus, saying "$K_n$ has $n$ vertices" is equivalent to saying "$K_n$ has $n$ vertices
for all $n$". Again, this is a claim about the whole collection $K_n$ for all $n$. Thus, "$K_n$ has 45 edges" is false just because there are elements of the collection with a differemt number of edges: e.g., $K_2$ has just 1 edge.
Joystar1977 said:
My understanding of this when it says the complete graph means the whole entire graph and I would think that if the graph has 5 vertices, denoted by K5 then Kn to my recollection is the unknown amount of edges in the graph.
"Complete" here is not used in its usual disctionary sense. A "complete graph" is an idivisible name, and it is given a new precise meaning by a definition. Namely, let $n$ be some natural number. Then a complete graph on $n$ vertices, denoted by $K_n$, consists of $n$ vertices, and every pair of distinct vertices is connected by an edge. This is also a whole series of definitions: for $n=1$, $n=2$ and so on. So, the word "complete" does not mean that some prevously defined graph is considered in its entirety; it is a part of a name that is given a new meaning. (Redefining familiar terms happens a lot in mathematics: we talk of "compact" and "meager" sets, properties holding "almost everywhere" or "eventually" and so on. Definitions give these terms new precise meanings different from those in the dictionary.) Similarly, $K_n$ is not a graph with an unknown number of edges. You can't talk about the number of edges in $K_n$ until you either give a concrete $n$ or use the phrases "for all $n$" or "for some $n$".
Joystar1977 said:
When I solve the problem you said to replace the first n (n - 1) by 5 and the second n by 10
Let's recall exactly:
Evgeny.Makarov said:
So, you replace the first $n$ in $n (n - 1)$ by 5, and you replace the second $n$ by 10. Do you think this is right?
I did not direct you to replace $n$ by two different numbers. I stated this as an observation and asked if you think it right. Have you ever seen the same variable in a single expression replaced by two values? I was hoping you notice this and feel ashamed because this is not done in mathematics.
Joystar1977 said:
My understanding of this when you say "Kn has 45 edges" is that "Kn is equal to having 45 edges".
In addition to the above, I want to stress that having 45 edges is not the same as being equal to 45 edges. Presumably, you have a computer, but you are not equal to a computer, right? I am not sure if the phrase "equal to having" is possible in English, but it is not correct here. "Having 45 edges" is at best a property of graphs, and a graph $K_n$ cannot be equal to a property, or a number, or a function and so on. A graph may only be equal to a graph. Theerefore, please don't say, "A graph is equal to 45 edges" or "A graph is equal to having 45 edges"; this is absolutely wrong.
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Joystar1977 said:
1. Define a graph.
A graph is a two-dimensional drawing showing a relationship usually between two sets of numbers by means of a line, curve, a series of bars, or other symbols. Usually, the independent variable is shown on the horizontal line (X-Axis) and the dependent variable is shown on the vertical line (Y-Axis).
No. Just no. Not in the context of this thread..
