Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286120210601A note on the first Zagreb index and coindex of graphs41511404710.22049/cco.2020.26809.1144ENIgorMilovanovićFaculty of Electronic Engineering, Nis, SerbiaMarjanMatejićFaculty of Electronic EngineeringEminaMilovanovićFaculty of Electronic EngineeringRanaKhoeilarAzarbaijan Shahid Madani University0000-0002-2981-3625Journal Article20200219Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, 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. http://comb-opt.azaruniv.ac.ir/article_14047_6dacca4d77087d8b3967a894b7a7d103.pdf