A note on the first Zagreb index and coindex of graphs

Let $G=(V,E)$, $V={v_1,v_2,ldots,v_n}$, be a simple graph with$n$ vertices, $m$ edges and a sequence of vertex degrees$Delta=d_1ge d_2ge cdots ge d_n=delta$, $d_i=d(v_i)$. Ifvertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $isim j$, otherwise, we write $insim j$. The first Zagreb index isvertex-degree-based graph invariant defined as$M_1(G)=sum_{i=1}^nd_i^2$, whereas the first Zagreb coindex isdefined as $overline{M}_1(G)=sum_{insim j}(d_i+d_j)$. A couple of new upper and lower bounds for $M_1(G)$, as well as a new upper boundfor $overline{M}_1(G)$, are obtained.