Degree-based function index of trees and unicyclic graphs

Source abstract

Abstract We use functions of two variables satisfying certain conditions to obtain graphs having the smallest value of the degree-based function index among trees and unicyclic graphs with given number of vertices. We show that those extremal results on trees and unicyclic graphs hold for many general degree-based indices such as the general reduced second Zagreb index $$GRM_{a}$$ G R M a for $$a \ge 0$$ a ≥ 0 , general Randić index $$R_{a}$$ R a for $$a > 0$$ a > 0 , first general Gourava index $$FGO_{a}$$ F G O a for $$a \ge 1$$ a ≥ 1 , general Z -type index $$Z_{a,b}$$ Z a , b for $$a \ge 1$$ a ≥ 1 , $$b \ge - 2$$ b ≥ - 2 , general Sombor index $$SO_{a,b}$$ S O a , b and one other generalization $$M_{a,b}$$ M a , b for $$a \ge 1$$ a ≥ 1 , $$b > 0$$ b > 0 . In the study of the maximum value for degree-based indices of trees with given number of vertices, we cover general indices such as $$SO_{a,b}$$ S O a , b for $$a > 0$$ a > 0 , $$b \ge 1$$ b ≥ 1 , and $$Z_{a,b}$$ Z a , b for $$a > 0$$ a > 0 , $$b \ge - 2$$ b ≥ - 2 .