VERTEX-BLOCK ZAGREB INDICES OF GRAPHS

A topological index is a graph invariant applicable in chemistry. The first and second Zagreb indices are topological indices based on the vertex degrees of molecular graphs. For any graph G, the first Zagreb index M\(G) is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices. A block is a maximal connected graph with no cut-vertices. The vertex-block degree (vb-degree) of a vertex is the number of blocks incident on it. In this paper, we define two new graph invariants, named as the first and second vertex-block Zagreb indices and obtain lower and upper bounds on them in terms of number of vertices, number of blocks and maximum vb-degree of a graph.