LAUSR

For a graph $$G = (V(G), E(G))$$G=(V(G),E(G)), let d(u), d(v) be the degrees of the vertices u, v in G. The first and second Zagreb indices of G are defined as $$ M_1(G) = \sum _{u \in V(G)} d(u)^2$$M1(G)=∑u∈V(G)d(u)2 and $$ M_2(G) = \sum _{uv \in E(G)} d(u)d(v)$$M2(G)=∑uv∈E(G)d(u)d(v), respectively. The first (generalized) and second Multiplicative Zagreb indices of G are defined as $$\Pi _{1,c}(G) = \prod _{v \in V(G)}d(v)^c$$Π1,c(G)=∏v∈V(G)d(v)c and $$\Pi _2(G) = \Pi _{uv \in E(G)} d(u)d(v)$$Π2(G)=Πuv∈E(G)d(u)d(v), respectively. The (Multiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. Denote by $${\mathcal {G}}_{n}$$Gn the set of all Eulerian graphs of order n. In this paper, we characterize Eulerian graphs with first three smallest and largest Zagreb indices and Multiplicative Zagreb indices in $${\mathcal {G}}_{n}$$Gn.