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Interval-valued dual hesitant fuzzy set (IVDHFS) as an extended structure of hesitant fuzzy set (HFS), interval-valued HFS and dual hesitant fuzzy set (DHFS) has been developed and applied in multi-attribute decision-making (MADM) problem. While deciding the membership degree (MD) and nonmembership degree (NMD) in q-rung orthopair fuzzy condition, decision makers perhaps hesitant among a lot of values, and the q-rung orthopair dual hesitant fuzzy sets (q-RODHFSs) were proposed. Decision makers prefer to utilize interval values, instated of crisp numbers, to represent MD and NMD in MADM problems. In this paper, we propose the novel idea of interval-valued q-rung orthopair dual hesitant fuzzy graphs, in the light of Hamacher operator, called interval-valued q-rung orthopair dual hesitant fuzzy Hamacher graphs (IV q-RODHFHGs) and determine its energy. Further, we develop the new concepts of Zagreb energy and Harmonic energy of IV q-RODHFHGs. Moreover, we apply the proposed concept of IV q-RODHFHGs to solve the MADM problems with IV q-RODHF information. Finally, a numerical model relating to the evaluation of the performance of search and rescue robots is presented to show the utilization of the developed technique. To interpret the effectiveness and the validity of the proposed method, a comparison analysis with the established approaches is conducted.