Extremal (n,m)-graphs with respect to distance-degree-based topological indices

In chemical graph theory, distance-degree-based topological indices are expressions of the form σ_u/v F(deg(u), deg(v), d(u, v)), where F is a function, deg(u) the degree of u, and d(u, v) the distance between u and v. Setting F to be (deg(u) + deg(v))d(u, v), deg(u)deg(v)d(u, v), (deg(u) + deg(v))d(u, v)^-1, and deg(u)deg(v)d(u, v)^-1, we get the degree distance index DD, the Gutman index Gut, the additively weighted Harary index HA, and the multiplicatively weighted Harary index HM, respectively. Let Gn,m be the set of connected graphs of order n and size m. It is proved that if G εG_n,m, where 4 ≤ n ≤ m ≤ 2n-4, then HA(G) ≤ (m(m+5)+2(n-1)(n-3))/2 and DD(G) ≤ (4m-n)(n-1)-(m-n+1)(m-n+6). The extremal graphs are characterized in both cases and are the same. Similarly, the graphs from Gn,m with m ≤ n + ^k ₂-k, 2 ≤ k ≤ n-1, maximizing the multiplicatively weighted Harary index and minimizing the Gutman index are obtained.