# Graph theory Lecture about opposite sex partners, I'm confused

This is a question about graph theory and understanding the math. I'm not interested in any politics or morality issues. I watched a lecture on Youtube from 2010 about graph theory. https://youtu.be/h9wxtqoa1jY

Around 5 minutes into the lecture he talks about 2 studies that found on average men have 74% and 233% more opposite sex partners than women. The lecturer drew a bipartite graph with men on the left and women on the right. He said he left out same sex relationships just because it would make the math more complicated.

He said Population in US at the time approx: |V| = 300 million, men |Vm| = 147.6 million , women |Vw| = 152.4 million

|Am| is average number of opposite sex partners for men = Sum of all degrees(Vm)/|Vm| which is equivalent to the sum of number of edges|E|/|Vm|. |Aw| is average number of opposite sex partners for women = |E|/|Vw|

so about 24 minutes in he writes Vm but says Vw and it gets confusing but I think he is saying: |Aw|/|Am| gives you Average number of partners women get per average that men get. and (|E|/|Vw|)/(|E|/|Vm|) can be simplified to |Vw|/|Vm|. Which = 1.0325

So I think he is saying that the studies were wrong, men get on average slightly more (approx 3%) partners than women because there are more women. But I'm not sure.

Also I was thinking that the studies seem to be about which sex prefers more partners. So for example if there was a room with 10 men and 10 women and 1 man had 10 women and the 10 women are not willing to have multiple partners then there are 9 men on their own not necessarily out of choice but just because theres not enough women in the room. So maybe the Vm with no edges should be removed from the equation so in this case Am= 10/1 = 10 and Aw 10/10 = 1. And this would give a better indication of how many partners someone of a certain sex would have if they had the opportunity.

P.S I'm not that good at maths, my logic, notation etc.. may be incorrect, I have many gaps but trying to improve :) Have a nice day.

Assuming "partner" is a symmetric relation, i.e. $A$ is a partner of $B$ iff $B$ is a partner of $A$, then if there are $E$ male-female partnerships, $V_m$ men and $V_w$ women, the average number of partners for a man is $E/V_m$ and the average number of partners for a woman is $E/V_w$. So if there are slightly more women, the average for a man is slightly more than the average for a woman.