Express the following statements using quantifiers and atomic statements How do i approach this problem?

Let G represent a graph with vertices V. The only atomic statements about the graph are of the form G(v,w) where v and w are vertices in V and G(v,w) means there is an edge between v and w. The quantifiers ∀x and ∃x refer to the vertex set V. With this context, express the following statements about the graph G formally using quantifiers and atomic statements:


There is some vertex in the graph that has an edge connecting it to every other vertex.
The graph is the complete graph.
Any two vertices are connected by a path of length 3.
The graph contains no triangle.
Among any four vertices at least two are connected by an edge.
I'll reword each of your three constraints in a way that's closer to the atomic-statement formulation:


There exists a vertex in the graph, such that for every other vertex, there is an edge between the two.
Any two vertices have an edge between them.
For any two vertices that we want to connect, we can find two new vertices to form a path between them (i.e. the first original vertex is connected to the first new vertex, the two new vertices are connected, etc.)


Comment if you need more guidance.
∃x:∀y:G(x,y)
∀x:∀y:G(x,y)
∀x:∀y:∃w:∃z:G(x,w)∧G(w,z)∧G(z,y)


The way I approach these types of problems is by translating them into constrained English ("for all x, ..." "there exists a y, ...") and then rendering them with equivalent formal notation.  For example, 1. can be translated as "there exists a vertex x such that for all vertices y, x and y are connected by an edge", and 2. can be translated as "for all vertices x and y, x and y are connected by an edge".  When I can't figure out how to translate them as such, that is a strong hint that either the problem is ill-specified or else my own understanding of the definitions is lacking.