On the Lanzhou Indices of Trees under Graph Decoration

The Lanzhou index of a graph G is defined as the sum of the product between and square of du over all vertices u of G, where du and are respectively the degree of u in G and the degree of u in the complement graph of G. R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. Lanzhou index is an important topological index. It is closely related to the forgotten index and first Zagreb index of graphs. In this note, we characterize the bound of Lanzhou index of R(T) of a tree T. And the corresponding extremal graphs are also determined.