how can we make 11 non-isomorphic graphs on 4-vertices?

Question

How can we draw all the non-isomorphic graphs on $4$ vertices ? But it is mentioned that $ 11 $ graphs are possible.

score 4 · Accepted Answer · edited Apr 13 '17 at 12:20

4

Just mentioning a couple of links you might find useful to answer similar questions.

edited Apr 13 '17 at 12:20

Community

1

answered Feb 20 '14 at 17:32

Ishfaaq

10,230

score 2 · Answer 2 · edited Dec 10 '17 at 08:46

2

Label the vertices $1,2,3,4$.

There are $11$ non-Isomorphic graphs.

With $0$ edges only $1$ graph
with $1$ edges only $1$ graph: e.g $(1,2)$ from $1$ to $2$
With $2$ edges $2$ graphs: e.g $(1,2)$ and $(2,3)$ or $(1,2)$ and $(3,4)$
With $3$ edges $3$ graphs: e.g $(1,2),(2,4)$ and $(2,3)$ or $(1,2),(2,3)$ and $(1,3)$ or $(1,2),(2,3)$ and $(3,4)$
with $4$ edges $2$ graphs: e.g $(1,2),(2,3),(3,4)$ and $(1,4)$ or $(1,2),(2,3),(1,3)$ and $(2,4)$
With $5$ edges only $1$ graph: $(1,2),(2,3),(3,4),(1,4)$ and $(1,3)$
With $6$ edges only $1$ graph: $(1,2),(2,3),(3,4),(1,4),(1,3)$ and $(2,4)$

All those non-isomorphic graphs are $1+1+2+3+2+1+1=11$

edited Dec 10 '17 at 08:46

JMP

22,637

answered Dec 10 '17 at 08:16

Victor Tomno

21

Thanks Victor Tomno! Your answer helped me correct my illustration - specifically, I referred to the number of simple graphs with 4 vertices with n edges from your post to correct my https://twitter.com/c010011012/status/1380804215900045313 while figuring out my self what those graphs looked like – cmarangu Apr 11 '21 at 05:51

score 2 · Answer 3 · answered Feb 20 '14 at 17:27

2

Start by drawing the 4 vertices. Then draw all the possible graphs with 0 edges (there is only one). Next, draw all the possible graphs with 1 edge (again, there is only one). Continue until you draw the complete graph on 4 vertices. You should end up with 11 graphs.

answered Feb 20 '14 at 17:27

D Wiggles

2,918

The part you describe as "Continue" is before enough information is available to establish the pattern which needs to be continued! It's like saying of the primes, start at 1, go to 2 and then so on! – Benjamin Feb 05 '16 at 20:52
We are ordering the graphs by the number of edges. First we draw all graphs with 0 edges, then 1, 2, $\ldots$, until we've made a complete graph (which has the maximal number of edges). – D Wiggles Feb 18 '16 at 23:36

score 0 · Answer 4 · answered Feb 20 '14 at 17:27

0

How many non-isomorphic graphs can you draw with $4$ vertices and $0$ edges? How many with $1$ edge? $2$? $\dots$

answered Feb 20 '14 at 17:27

Michael Biro

13,847

with 4 vertices all graphs drawn are isomorphic if the no. of edges are 0,1,2. – mahavir Feb 22 '14 at 03:14
@mahavir This is not true with 4 vertices and 2 edges. – Sep 15 '17 at 16:56

how can we make 11 non-isomorphic graphs on 4-vertices?

4 Answers4