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Abstract The weighted vertex PI index of a graph G is defined by P I w (G) = ∑ e = u v ∈ E (G) (d G (u) + d G (v)) (n u (e | G) + n v (e | G)) where n u (e | G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. In this paper, we give the upper bound and the corresponding extremal graphs on the weighted vertex PI index of (n, m)-graphs with diameter d. The lower bound and the corresponding extremal graphs on the first Zagreb index and the weighted vertex PI index of trees with diameter d are given by two procedures. The extremal graphs, given by the two procedures, are also the extremal graphs which attain the lower bound on the first Zagreb index among all connected graphs with n vertices and diameter d. [ABSTRACT FROM AUTHOR]

For a graph G containing no component isomorphic to the 2-vertex path graph, the augmented Zagreb index (AZI) of G is defined as A Z I (G) = ∑ u v ∈ E (G) (d (u) d (v) d (u) + d (v) − 2) 3. This topological index has been proved to be closely correlated with the formation heat of heptanes and octanes. A k -apex tree is a connected graph G admitting a k -subset of vertices X such that G − X is a tree, but for any subset of vertices X ′ of order less than k , G − X ′ is not a tree. In this paper, we determine the minimum AZI among all k -apex trees for k ∈ { 1 , 2 , 3 }. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be an n -vertex graph, where V = { v 0 , v 1 , ... , v n − 1 }. The augmented Zagreb index (A Z I) of G is defined as A Z I (G) = ∑ v i v j ∈ E [ d i d j / (d i + d j − 2) ] 3 , where d i is the degree of v i. Let T n d be the set of all trees on n vertices with given diameter d. In this paper, we determine the tree with maximum A Z I among T n d when n ≥ 3 2 (d − 1) + 381. Our result partially resolve a problem given in [12]. [ABSTRACT FROM AUTHOR]

• The effect of graft transformations to decrease and/or increase the Zagreb eccentricity indices is studied. • Sharp lower bounds on Zagreb eccentricity indices of graphs are established in C (n , k) , where C (n , k) is the class of all cacti of order n with k cycles. • Sharp upper bounds on Zagreb eccentricity indices of graphs are established in C (n , k) , where 0 ≤ k ≤ ⌊ n − 1 2 ⌋. • Corresponding extremal graphs are characterized. For a connected graph G , the first Zagreb eccentricity index is defined as ξ 1 (G) = ∑ u ∈ V (G) e 2 (u) , and the second Zagreb eccentricity index is defined as ξ 2 (G) = ∑ u v ∈ E (G) e (u) e (v) , where e (u) is the eccentricity of u in G. Let C (n , k) be the class of all cacti of order n with k cycles. In this paper, we establish sharp lower bounds on Zagreb eccentricity indices of graphs in C (n , k) and determine the corresponding extremal graphs. What's more, we characterize the graphs in C (n , k) with maximal Zagreb eccentricity indices, where 0 ≤ k ≤ ⌊ n − 1 2 ⌋. [ABSTRACT FROM AUTHOR]