On the vv-degree based first Zagreb index of graphs

A topological index is a graph invariant applicable in chemistry. The first Zagreb index is a topological index based on the vertex degrees of molecular graphs. For any graph G, the first Zagreb index (Formula presented.) is equal to the sum of squares of the degrees of vertices. A block in a graph G is a maximal connected subgraph of G which has no cut-vertices. Two vertices (Formula presented.) are said to be vv-adjacent if they incident on the same block. The vv-degree of a vertex u is the number of vertices vv-adjacent to u. In this paper, we introduce a vv-degree based graph invariant, named the first vv-Zagreb index (Formula presented.), and obtain lower and upper bounds on (Formula presented.) in terms of the number of vertices, number of blocks, and maximum vv-degree of G using some classical inequalities. Further, we compute the first vv-block Zagreb index for the silicate network and the silicate chain network.