| Summary: | In this paper the problem of minimizing vertex-degree function index Hf(G) for kgeneralized quasi-unicyclic graphs of order n is solved for k ≥ 1 and n ≥ 2k + 2 if the function f is strictly increasing and strictly convex. These conditions are fulfilled by general first Zagreb index 0Rα(G) if α > 1, second multiplicative Zagreb index (Formula Presented) and sum lordeg index SL(G). The extremal graph is unique for k = 1, n = 4 and for k ≥ 2 2 and it consists from a path x1, x2, . . ., xn−1 and a new vertex xn adjacent with xk, xk+1 and xk+2,. © 2022 Art of Discrete and Applied Mathematics. All rights reserved.
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