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

الأصل · EN

Let TΔₙ denote the set of trees of order n, in which the degree of each vertex is bounded by some integer Δ. Suppose that every tree in TΔₙ is equally likely. We show that the number of vertices of degree j in TΔₙ is asymptotically normal with mean (μⱼ+o(1))n and variance (σⱼ+o(1))n, where μⱼ, σⱼ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in TΔₙ. Moreover, we obtain that the number of edges of type (i,j) in TΔₙ also has mean (μij+o(1))n and variance (σij+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 μij, σij are some constants. Then, we give estimate to the value of the general Randić index for almost all trees in TΔₙ.