Abstract
For a graph G, let σ(G)=∑uv∈E(G)1dG(u)+dG(v); this defines the sum-connectivity indexσ(G). More generally, given a positive function t, the edge-weightt-indext(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 12nkt(2k), with equality only for k-regular graphs.
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