First zagreb index, κ-connectivity, β-deficiency and κ-hamiltonicity of graphs

The first Zagreb index M₁(G) is equal to the sum of squares of the degrees of the vertices of the underlying molecular graph G: Finding suffcient conditions for graphs possessing certain properties are important problems. In this paper, we give suffcient conditions in terms of the first Zagreb index for a graph to be κ-connected, β-deficient or κ-hamiltonian.