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An \( (a, d) \)-edge-antimagic total labeling of a graph \( G \) with \( p \) vertices and \( q \) edges is a bijection \( f \) from the set of all vertices and edges to the set of positive integers \( \{1, 2, 3, \dots, p+q\} \) such that all the edge-weights \( w(uv) = f(u) + f(v) + f(uv) \) for \( uv \in E(G) \), form an arithmetic progression starting from \( a \) and having common difference \( d \). An \( (a, d) \)-edge-antimagic total labeling is called a super \( (a, d) \)-edge-antimagic total labeling (\((a,d)\)-SEAMT labeling) if \( f(V(G)) = \{1, 2, 3, \dots, p\} \). The graph \( F_n \), consisting of \( n \) triangles with a common vertex, is called the friendship graph. The generalized friendship graph \( F_{m_1, m_2, \dots, m_n} \) consists of \( n \) cycles of orders \( m_1 \leq m_2 \leq \dots \leq m_n \) having a common vertex. In this paper, we prove that the friendship graph \( F_{16} \) does not admit a \( (a, 2) \)-SEAMT labeling. We also investigate the existence of \( (a, d) \)-SEAMT labeling for several classes of generalized friendship graphs.