The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degree

Abstract

Let ${\mathcal{T}}_n^{\Delta }$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\Delta$. Suppose that every tree in ${\mathcal{T}}_n^{\Delta }$ is equally likely. We show that the number of vertices of degree $j$ in ${\mathcal{T}}_n^{\Delta }$ is asymptotically normal with mean $({\mu }_j+o(1))n$ and variance $({\sigma }_j+o(1))n$, where ${\mu }_j$, ${\sigma }_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in ${\mathcal{T}}_n^{\Delta }$. Moreover, we obtain that the number of edges of type $(i,j)$ in ${\mathcal{T}}_n^{\Delta }$ also has mean $({\mu }_{ij}+o(1))n$ and variance $({\sigma }_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and ${\mu }_{ij}$, ${\sigma }_{ij}$ are some constants. Then, we give estimate to the value of the general Randić index for almost all trees in ${\mathcal{T}}_n^{\Delta }$.

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