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