Number of Subgraphs of Kn?

Question

I was searching for this question and found below link

How many subgraphs does $K_5$ have?

Here, in general, it is given to be

$\displaystyle\sum_{r=0}^{n}\binom{n}{r}2^{\frac{r(r-1)}{2}}$

My doubt is that the case for $r=0$ is taken if we select none of the vertices of $K_n$?

Why the case for $ r=0$ considered?

Comments

@Ayush
is it leveled graph or unleveled?

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this formula is for leveled graph

also we can do it manually

right?

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@srestha-it is for both.

Answer 1

but according to definition of graph it must have atleast one vertex otherwise there is no graph-reference(narsingh deo)

These are things where there is no Universally accepted definitions. If you refer different text books then you will find different conventions. That's why when we read any books...Author first defines certain things and  then builds concepts around it. Graph theory is one of most popular example of what I just said.

Like take for instance...What is the definition of Walk, Path, Trail, Multi graph etc...and many more things. 

The formula that you mentioned for Number of subgraphs of $K_n$ assumes that No vertex at all is also one kind of graph.. may be called Null graph or empty  graph (However Author has termed it)... Moreover, This formula is for Labelled Graph. i.e. Every vertex forms different subgraph.

Comments

And also reading from 1 book maynot be sufficient

Answer 2

The answer simply counts the case of $null graph$( a graph without vertices). Whether a subgraph definition allows this is debatable. Here's a post you should read.