Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs

For a graph \(G = (V(G), E(G))\), 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 \( M_1(G) = \sum _{u \in V(G)} d(u)^2\) and \( M_2(G) = \sum _{uv \in E(G)} d(u)d(v)\), respectively. The first (generalized) and second Multiplicative Zagreb indices of G are defined as \(\Pi _{1,c}(G) = \prod _{v \in V(G)}d(v)^c\) and \(\Pi _2(G) = \Pi _{uv \in E(G)} d(u)d(v)\), 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 \({\mathcal {G}}_{n}\) 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 \({\mathcal {G}}_{n}\).