# Graph Theory: A has 15 edges, Ā has 13 edges, how many vertices does A have?

• Jun 22nd 2008, 08:55 PM
yvonnehr
Graph Theory: A has 15 edges, Ā has 13 edges, how many vertices does A have?
Discrete Math: Graph Theory #54.
If A is a simple graph with 15 edges, and Ā has 13 edges, how many vertices does A have?

Can anyone give me the answer and how to arrive at it?

Thanks a bunch,
Yvonne(Sleepy)
• Jun 22nd 2008, 09:40 PM
kalagota
Quote:

Originally Posted by yvonnehr
Discrete Math: Graph Theory #54.
If A is a simple graph with 15 edges, and Ā has 13 edges, how many vertices does A have?

Can anyone give me the answer and how to arrive at it?

Thanks a bunch,
Yvonne(Sleepy)

do you know this statement?

Let $\displaystyle \overline{G}$ be the complementary graph of $\displaystyle G$. Then
$\displaystyle |E(\overline{G})| + |E(G)| = \left({\begin{array}{c} |V(G)| \\ 2 \end{array}}\right)$
• Jun 22nd 2008, 09:48 PM
yvonnehr
Equation -> Handshaking Theorem?
Quote:

Originally Posted by kalagota
do you know this statement?

Let $\displaystyle \overline{G}$ be the complementary graph of $\displaystyle G$. Then
$\displaystyle |E(\overline{G})| + |E(G)| = \left({\begin{array}{c} |V(G)| \\ 2 \end{array}}\right)$

I don't believe I have seen this. Is this related or some rendition of the Handshaking Theorem?
-Ivy
• Jun 22nd 2008, 09:59 PM
kalagota
i dont think so.. we discussed this as a remark in graph operations..
• Jun 22nd 2008, 10:05 PM
yvonnehr
Quote:

Originally Posted by kalagota
do you know this statement?

Let $\displaystyle \overline{G}$ be the complementary graph of $\displaystyle G$. Then
$\displaystyle |E(\overline{G})| + |E(G)| = \left({\begin{array}{c} |V(G)| \\ 2 \end{array}}\right)$

Tell me I am reading this right.

The sum of the cardinality of (edges) G and G-compliment equals to the cardinality of (vertices) G.

Sorry if I am not getting this correctly. It is very late for me. :-) You know how it goes.
• Jun 22nd 2008, 10:20 PM
kalagota
Quote:

Originally Posted by yvonnehr
Tell me I am reading this right.

The sum of the cardinality of (edges) G and G-compliment equals to the cardinality of (vertices) G.

Sorry if I am not getting this correctly. It is very late for me. :-) You know how it goes.

... cardinality of (vertices) G taken 2..

EDIT: that is the usual combination formula..