Extremal problems for degree-based topological indices

For a graph G, let σ(G) = ∑ uv∈E(G) 1 √ dG(u)+dG(v) ; this defines the sum-connectivity index σ(G). More generally, given a positive function t, the edge-weight t-index t(G) is given by t(G) = ∑ uv∈E(G) t(ω(uv)), where ω(uv) = dG(u) + dG(v). We consider extremal problems for the t-index over various families of graphs. The sum-connectivity index satisfies the conditions imposed on t in each extremal problem, with a small exception. Minimization: When t is decreasing, and (z − 1)t(z) is increasing and subadditive, the star is the unique graph minimizing the t-index over n-vertex graphs with no isolated vertices. When also t has positive second derivative and negative third derivative, and (z − 1)t(z) is strictly concave, the connected n-vertex non-tree with least t-index is obtained from the star by adding one edge. Maximization: When t is decreasing, convex, and satisfies t(3)− t(4) < t(4)− t(6), the path and cycle are the unique n-vertex tree and unicyclic graph with largest t-index. When also t(4)− t(5) ≤ 2[t(6)− t(7)], and t(k+1)− t(k+2)− t(k+ j) increases with k for j ≤ 3, we determine the n-vertex quasi-trees with largest t-index, where a quasi-tree is a graph yielding a tree by deleting one vertex. The maximizing quasi-trees consist of an n-cycle plus chords from one vertex to some number c of consecutive vertices (for the sum-connectivity index, c = min{30, n − 3}). Finally, we show that whenever t is decreasing and zt(z) is strictly increasing, an n-vertex graph with maximum degree k has t-index at most 1 2nkt(2k), with equality only for k-regular graphs.