Write these graph properties as first order expression (assume graph is undirected)

Question

1. The given graph is a clique (there is edge between every pair of vertices)

2. Every vertex has degree at least two (there is minimum two edges starting from every vertex )

Answer 1

$$\begin{align} E(x,y) \quad &=\quad \substack{\large\text{There is an edge between}\\\large\text{the vertecies $x$ and $y$}\\\large\text{in the given graph}}\\[3em] \substack{\large\text{The given graph}\\\large G \text{ is a clique}} \quad &= \quad \forall x,y: \left ( \substack{\large\quad x \in G\\\large\land\;\, y \in G\\\large \land \;\, x \neq y} \implies E(x,y)\right )\\[3em] \substack{\large\text{Every vertex in } G\\\large \text{has degree }\geq 2} \quad &= \quad \forall x \left (x \in G \implies  \exists v\;\exists w: \left ( \substack{\large E(x,v)\land E(x,w)\\\large \land\; x\neq v\; \land \; x\neq w\\\large \land\; v \neq w} \right )\right ) \end{align}$$