The first and the second Zagreb indices of the generalized Mycielskian of graphs

The first and the second Zagreb indices of a graph G   are defined as M1(G)=∑v∈V(G)(dG(v))2M1(G)=∑v∈V(G)(dG(v))2 and M2(G)=∑uv∈E(G)(dG(u)dG(v))M2(G)=∑uv∈E(G)(dG(u)dG(v)) respectively, where dG(u)dG(u) denotes the degree of the vertex u in G. In this work, we compute the first and the second Zagreb indices of the generalized Mycielskian of a graph G  , denoted by μk(G)μk(G) and the complement of μk(G)μk(G), denoted by μk(G)‾ in terms of the order and size of the graph G. Also, we obtain exact expressions for the first and the second Zagreb indices of the generalized Mycielskian of some graph operations and sharp upper bounds for the first and the second Zagreb indices of the generalized Mycielskian of graphs.