**Abstract: ** The Zagreb indices have been introduced more than forty four years ago by Gutman and
Trinajestic as the sum of the squares of the degrees of the vertices, and the sum of the products of the degrees of pairs of adjacent
vertices, respectively, \cite{za1}. In this paper, we introduce the first and second equitable and non-equitable Zagreb indices as
$M_1^e(G)=\sum_{u\in V(G)}\big[deg_e(u)\big]^2$, $M_2^e(G)=\sum_{uv\in E(G)}deg_e(u)deg_e(v)$,
$M_1^{ne}(G)=\sum_{u\in V(G)}\big[deg_{ne}(u)\big]^2$ and $M_2^{ne}(G)=\sum_{uv\in E(G)}deg_{ne}(u)deg_{ne}(v)$,
respectively, where $deg_e(u)$ and $deg_{ne}(u)$ denotes the equitable and non-equitable degrees of vertex $u$. Exact values for wheel,
firecracker and firefly graph families are obtained, some properties of the equitable and non-equitable Zagreb indices are established.

**Keywords: ** First equitable Zagreb index, Second equitable Zagreb index, First non-equitable Zagreb index, Second non-equitable Zagreb index.

Akram Alqesmah

Department of Studies in Mathematics, University of Mysore, Mysore, India.

E-mail: aalqesmah@gmail.com