On the maximally edge-connected square graphs with degree sequence conditions

Let $G=(V(G),E(G))$ be a finite connected simple graph. The edge-connectivity $\lambda (G)$ of $G$ is the minimum number of edges that must be removed to make $G$ disconnected. We call $G$ maximally edge-connected when $\lambda (G)=\delta (G)$, where $\delta (G)$ is the minimum degree of $G$. The square graph $G^2$ of $G$ is the graph with vertex set $V(G)$, where two distinct vertices $u$ and $v$ are adjacent in $G^2$ precisely when their distance in $G$ is at most two. In this paper, first we prove that if $\lambda (G)\ge 4$ and $\lambda (G^2)<\delta (G^2)$, then $\lambda (G^2)>\lambda (G{)}^{3\mathrm{/}2}-\frac{5}{4}\lambda (G)$, which extends the result in \cite{ChenXu} from the edge-connectivity range $4\le \lambda (G)\le 7$ to $\lambda (G)\ge 4$. Then we show that $G^2$ is maximally edge-connected if one of the following conditions hold: (i) $\delta (G)\ge \lfloor (n+2)\mathrm{/}4\rfloor$, or $\delta (G)\le \lfloor (n+2)\mathrm{/}4\rfloor -1$ and ${\sum }_{i=1}^{2k}{d}_{n+1-i}\ge k(n-4)-2\delta (G)+3$, where $k$ is an integer satisfying $2\le k\le \delta (G)+2$ and $d_1\ge d_2\ge \cdots \ge d_n$ are the degrees of $G$; (ii) $d_{n-\delta (G)}+d_n\ge (3n-9)\mathrm{/}4$.