Degree-based function index of trees and unicyclic graphs

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2024年11月28日

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Tomáš Vetrík

摘要

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 $a \ge 0$ $a \geq 0$ , general Randić index $R_{a}$ $R_{a}$ for $a > 0$ $a > 0$ , first general Gourava index $FGO_{a}$ $F G O_{a}$ for $a \ge 1$ $a \geq 1$ , general Z-type index $Z_{a,b}$ $Z_{a, b}$ for $a \ge 1$ $a \geq 1$ , $b \ge - 2$ $b \geq - 2$ , general Sombor index $SO_{a,b}$ $S O_{a, b}$ and one other generalization $M_{a,b}$ $M_{a, b}$ for $a \ge 1$ $a \geq 1$ , $b > 0$ $b > 0$ . 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 $a > 0$ $a > 0$ , $b \ge 1$ $b \geq 1$ , and $Z_{a,b}$ $Z_{a, b}$ for $a > 0$ $a > 0$ , $b \ge - 2$ $b \geq - 2$ .