a note on the first zagreb index and coindex of graphs

Let g=(v,e), v={v1,v2,…,vn}, be a simple graph with n vertices, m edges and a sequence of vertex degrees δ=d1≥d2≥⋯≥dn=δ, di=d(vi). if vertices vi and vj are adjacent in g, it is denoted as i∼j, otherwise, we write i≁j. the first zagreb index is vertex-degree-based graph invariant defined as m1(g)=∑^n i=1d^2i, whereas the first zagreb coindex is defined as m¯¯¯¯¯1(g)=∑i≁jdi+dj). a couple of new upper and lower bounds for m1(g), as well as a new upper bound for m¯¯¯¯¯1(g), are obtained.