Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 6 1 2021 06 01 A note on the first Zagreb index and coindex of graphs 41 51 14047 10.22049/cco.2020.26809.1144 EN Igor Milovanović Faculty of Electronic Engineering, Nis, Serbia Marjan Matejić Faculty of Electronic Engineering Emina Milovanović Faculty of Electronic Engineering Rana Khoeilar Azarbaijan Shahid Madani University Journal Article 2020 02 19 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 $\Delta=d_1\ge d_2\ge \cdots \ge d_n=\delta$, $d_i=d(v_i)$. If vertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $i\sim j$, otherwise, we write $i\nsim j$. The first Zagreb index is vertex-degree-based graph invariant defined as $M_1(G)=\sum_{i=1}^nd_i^2$, whereas the first Zagreb coindex is defined as $\overline{M}_1(G)=\sum_{i\nsim j} d_i+d_j)$. A couple of new upper and lower bounds for $M_1(G)$, as well as a new upper bound for $\overline{M}_1(G)$, are obtained.  https://comb-opt.azaruniv.ac.ir/article_14047_6dacca4d77087d8b3967a894b7a7d103.pdf