if G=(A,B,E) is a graph without directions. and (i mean unite value) |A U B| =50 what is the max value of |E| ?
im very new at the topic and if there is anywebsite that also can explain more.. it will be very helpfull... thanks
WHAT I DID: (probally its worng...) if |A U B| =50 then max value of |A|=|B|=25 for each what means the |E|=625 hopefully it correct?
It seems that you mean that $G$ is an undirected bipartite graph with partite sets of vertices $A$ and $B$ and edge set $E$, with $|A \cup B| = 50$. So, you are asking for the maximum number of edges in a bipartite graph with 50 vertices.
The number of edges in a complete bipartite graph whose partite sets have size $m$ and $n$ respectively is $mn$. (Every edge contains exactly one vertex in the partite set of size $m$; for each of these $m$ vertices, there are edges to each of the $n$ vertices on the other side.)
And, given any bipartite graph $G$, obviously the complete bipartite graph on the same partite sets has at least as many edges as $G$. So your question boils down to finding $n$ between 0 and 50 so that $n(50-n)$ is maximized.
One way to do this is calculus: to find the maximum of $50n - n^2$, solve for its derivative to be 0: $$50 - 2n = 0$$ has the solution $n = 25$. This critical point is indeed a maximum (take the second derivative if you want to check).
Then the number of edges is $25 \cdot 25 = 625$.