Riktade grafer - Diskret matematik

Hej! Jag har lite svårt för det här med grafer och har en uppgift jag suttit med väldigt länge nu men kommer inte riktigt någonvart...

Jag tror att svaret är (n² - n) /2 men jag vet inte hur jag ska bevisa det.

For a directed graph ("riktad graf") G with n nodes, let A(n) be the maximum number of arrows ("pilar", "riktade kanter") G can have, without G having a cycle.

Your job is to investigate what A(n) is.

Examples:

A directed graph with 2 nodes can have 1 arrow without there being a cycle, but it must have a cycle with more than 1 arrow. So A(2) = 1.
A directed graph with 3 nodes can have 3 arrows without there being a cycle, but it must have a cycle with more than 3 arrows. So A(3) = 3.
A directed graph with 4 nodes can have 6 arrows without there being a cycle, but it must have a cycle with more than 6 arrows. So A(4) = 6.
Express your answer A(n) in terms of n. For your answer, explain both:

(a) that there exist directed graphs with n nodes that have A(n) arrows and no cycles, and

(b) that directed graphs with n nodes with more arrows than A(n) must have a cycle.