Further Properties of the Vertex-Block Zagreb Indices of Graphs

A topological index is a graph invariant with applications in chemistry. The first and second Zagreb indices are degree-based topological indices of molecular graphs. A block of a graph G is a maximal connected subgraph of G that contains no cut-vertex. The vb-degree of a vertex is defined as the number of blocks incident to it. The first and second vertex-block Zagreb indices, denoted by VBM 1 (G) and VBM 2 (G), respectively, are vb-degree-based topological indices. In this paper, we determine the graphs that attain the maximum and minimum values of the first and second vertex-block Zagreb indices among all graphs with a given number of vertices and blocks. Furthermore, we establish lower and upper bounds for VBM 1 (G) and VBM 2 (G) in terms of the number of vertices, the number of blocks, the maximum block order, and the maximum vb-degree of a graph. We also characterize the extremal graphs attaining these bounds.