On the vv-degree based first Zagreb index of graphs

Abstract

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: see text] 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: see text] 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: see text], and obtain lower and upper bounds on [Formula: see text] 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.

Keywords

• First Zagreb index
• vv-degree
• block Zagreb indices
• first vv-Zagreb index
• 05C07
• 05C09