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Mathematical analysis reveals the maximal first Zagreb index for cactus graphs of fixed order and Roman domination number, characterizing all corresponding extremal graph structures.
The first Zagreb index (M1-index) of a graph G is defined as the sum of the squares of the vertex degrees, which is one of the oldest and most thoroughly studied degree-based topological indices. A cactus is a connected graph and all of its blocks are either edges or cycles. In this paper, we determine the maximum first Zagreb index of cacti with given order and Roman domination number, and characterize all corresponding extremal graphs.
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